Deviation from Linear Elastic Fracture in Near-Surface Hydraulic Fracturing Experiments with Rock Makhnenko, R.Y.

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1 ARMA Deviation from Linear Elastic Fracture in Near-Surface Hydraulic Fracturing Experiments wit Rock Maknenko, R.Y. University of Minnesota, Minneapolis, MN, USA Bunger, A.P. CSIRO Eart Science and Resource Engineering, Clayton, Victoria, Australia Detournay, E. University of Minnesota, Minneapolis, MN, USA Copyrigt 2010 ARMA, American Rock Mecanics Association Tis paper was prepared for presentation at te 44 t US Rock Mecanics Symposium and 5 t U.S.-Canada Rock Mecanics Symposium, eld in Salt Lake City, UT June 27 30, Tis paper was selected for presentation at te symposium by an ARMA Tecnical Program Committee based on a tecnical and critical review of te paper by a minimum of two tecnical reviewers. Te material, as presented, does not necessarily reflect any position of ARMA, its officers, or members. Electronic reproduction, distribution, or storage of any part of tis paper for commercial purposes witout te written consent of ARMA is proibited. Permission to reproduce in print is restricted to an abstract of not more tan 300 words; illustrations may not be copied. Te ABSTRACT. Tis paper presents te results of near-surface ydraulic fracturing experiments in a medium-grained gabbro tat make use of different specimen sizes in order to observe te influence of rock eterogeneity and/or non-lefm beavior on fracture propagation. A comparison of te results wit laboratory experiments in brittle elastic materials sows tat fracture pats in rock usually do not monotonically approac te free surface. Moreover, tey exibit a ig degree of non-symmetry, in contrast to experiments in glass or PMMA. An attempt to correlate crack pat variation among different sized specimens to a size dependent fracture tougness as been initially inconclusive due to te noise caused by tese crack pat complexities. Noneteless, tese results provide an initial demonstration of non-lefm beavior in ydraulic fracturing wit implications for ongoing experiments aimed at quantifying te size effect on rock fracture tougness. 1. INTRODUCTION Laboratory tests indicate dependence of rock strengt on specimen size. In particular, te nominal tensile stress associated wit specimen failure decreases in an empirical power-law relationsip to specimen size (e.g. [1]). Tis is usually explained by reasoning tat tensile strengt is not a true material property but depends on te specimen geometry and probable maximum flaw size tat is subjected to tensile loading. Anoter manifestation of te size effect is expressed in te critical energy release rate associated wit fracture extension fracture energy, for sort, wic relates to te square of te fracture tougness increasing usually in an empirical power-law relationsip to specimen/notc/intact ligament/crack size (e.g. [2,3]). Altoug previous experimental works ave indicated in most cases a power-law relationsip at te laboratory scale ( < 1 m), and ave led to a number of teoretical recommendations regarding te treatment of fracture in eterogeneous materials (e.g. [4,5,6,2]), ow tese results and teories can be appropriately extended to larger scales remains fundamentally unclear. Te overarcing objective of tis work is to develop a new metod for estimating ow te fracture tougness depends on te size of te fracture tat is practically applicable at scales far exceeding laboratory limitations. It is based on ydraulically fracturing rocks at a sallow dept, tat is, so tat te radius of an initially orizontal, circular fracture attains a value tat is several times greater tan its initial dept. As tese sallow ydraulic fractures grow, te asymmetry of te fracture opening induces curving of te fracture towards te surface so tat te geometry resembles a concave-upward saucer. Previous researc as sown tat te saucer-sapes are, for omogeneous brittle materials suc as glass and polymetyl metacrylate (PMMA), determined to leading order by a parameter tat compares te in-situ compressive stress acting parallel to te free surface wit a fracture tougness dependent estimate of te magnitude of te fracture-induced stress field [7,8,9]. Hence, tese prior works sow tat te final saucersape relates directly to te fracture tougness of te material. Terefore, wit knowledge of te in-situ stress field, one may in principle estimate te fracture tougness based on a posteriori measurements of te fracture sape and witout requiring typically unreliable

2 measurement or analysis of te internal loading. Furtermore, tis metod may be applied at virtually any scale, provided one can measure te final fracture surface. If te experiments in omogeneous brittle materials, referenced above, comprise a first step in development of tis metod, tis paper comprises a second step. Tis step is to perform laboratory scale experiments in differently sized rock specimens to determine if tere is evidence tat te fracture tougness is a function of te specimen size at te laboratory scale, and more broadly, to determine te extent to wic growt of small scale ydraulic fractures in rocks compares wit omogeneous brittle elastic experiments. were! " # corresponds to te case were te crack grows parallel to te surface. Te validity of! as te parameter controlling te saucer sape as been demonstrated experimentally for omogeneous, brittle elastic materials (crack pats for different values of! are sown in Figure 1) [7,8,9]. Tat is, for experiments in different materials and using differently-sized specimens, te saucer sape was found to be te same, up to a rescaling of te radial and dept coordinates by te initial dept H, provided tat! was te same from one experiment to te next. 2. THEORETICAL BACKGROUND Wen ydraulic fractures grow near a free surface of a alf-space tat is not subjected to far field (or in te laboratory, applied) stresses, te fracture curves toward te surface to form a concave upward saucer werein te radial distance from te injection point to te point at wic te crack intersects te free surface, te dayligt radius R D, is between 3 and 4 times te initial dept H (e.g. [7,8,9,10]). It is te asymmetry in compliance between te two sides of te fracture tat ultimately causes te crack pat to curve towards te surface. Existence of far field stress! acting parallel to te initial plane of fracture growt reduces te fracture curving towards te free surface. In fact if! is muc larger tan any oter stress relevant to te crack growt, te optimum crack pat will be orizontal in order to maintain te crack opening in te direction of te least compressive stress. Te crack pat, and terefore te final sape of te saucer, is tus te result of an interaction between te crack s desire to curve, due to its asymmetry, and to remain parallel to te free surface in order to remain properly oriented relative to te far-field stress. As originally suggested by Zang et al. [11], tis competition can be quantified via a dimensionless parameter tat compares! wit te magnitude of te crack-induced, asymmetric stress field. In te context of Linear Elastic Fracture Mecanics (LEFM), te latter stress is classically estimated as K / R, were K is te opening mode fracture tougness and R is te crack lengt or radius. For a sallow ydraulic fracture, R! H, were H is te initial dept. Hence, te socalled curving parameter! is given by! K " = (1) H Fig. 1. Crack pats for experiments performed in PMMA and glass [8]. Hence,! is a similarity parameter [12] governing saucer-saped ydraulic fracture growt. However,! is only a similarity parameter if K is a true material parameter, not varying wit te size of te experiment or, for tat matter, te stress state. But it is well known tat for rocks K is indeed a function of te crack lengt. We arrive, ten, to te working ypotesis underlying tese experiments: For te same rock and te same nominal value of! based on a single measured K, te saucer-sape resulting from sallow ydraulic fracture growt is expected to be flatter te smaller te experimental scale (i.e. for smaller H ). Tis dependence of te saucer sape on te size would reflect te fact tat te effective K is expected to be smaller in te smaller experiments, ence! would be larger and te saucer sape flatter. Te experiments described below are designed as an initial, laboratory-scale test of tis ypotesis. 3. EXPERIMENTAL OVERVIEW Saucer-saped ydraulic fractures were created in blocksaped specimens of a medium-grained (~2 mm grain size) gabbro, commercially known as Adelaide Black Granite, by injecting glycerin solutions at a constant rate using a positive displacement stepping motor pump, as sown in Figure 2. Te fractures were initiated from a

3 Fig 2. Cross-section of te setup for te laboratory experiments mm deep circumferential notc at te base of a ole tat terminated a prescribed distance from te blocks surface. In eac case, a stainless steel tube was epoxy cemented into te ole, connected to te pump, and tus used to deliver te fluid to te notc. Upon commencing pumping, te pressure would increase to breakdown, after wic te ydraulic fracture would grow from te notc wile te injection pressure decreased. Finally, te fracture reaced eiter te bottom surface or one of te sides of te block, at wic point te pumping ceased and te test was completed.. Tree main design criteria were used. Firstly, te experiments were designed so tat te average or maximum ydraulic fracture radius attained in a test, wic is a function of te initial dept H (Figure 2), would vary relative to te grain size in te rock. For example, Mulmule and Dempsey [13] observed a size effect on K in sea ice wen te crack size was less tan about 200 times te largest grain size. In our case, investigation of te effect of varying of te crack size relative to te grain size makes use of two block sizes: small (200! 200! 120 mm, H = 22 mm, ole diameter 8.6 mm) and medium (400! 400! 235 mm, H = 45 mm, ole diameter 18.3 mm). Hence te maximum crack radius is circa 50 times te grain size for te small blocks and 100 times te grain size in te large blocks. Te second experimental design criterion is to ensure socalled pysical similarity [12], by keeping te parameter! from equation (1) te same between comparable tests in te small and medium blocks. Here! is te orizontal stress applied using water-filled stainless steel flat jacks and K is te fracture tougness. Note tat te nominal value for K was found using semi-circular beam tests [14] to be in te range of 1.27! 2.98 MPa m, increasing wit specimen size, wit te notc lengt ranging from mm. Finally, we wanted to ensure tat te influence of viscous flow can be neglected, suc tat all te experiments were run in a so-called tougnessdominated regime, wic requires te experimental duration to far exceed a caracteristic time associated wit te fracture s transition from te viscosity to tougness dominated regime, given by [15] µ " Q t mk! E! % $ ' (2) # & 18 K For tese experiments, 0.79! Q 0! 3.0 mm 3 / sec is te constant injection rate, 1/2 2 E" = E / (1 #! ) for Young s modulus E = 91GPa and Poisson s ratio! = 0.30, and µ = 0.83Pa!sec is te dynamic viscosity of te glycerin solution at te usual experimental temperature of 19 0 C. Hence, we obtain t < sec for tese experiments tat always lasted mk more tan 5 seconds, wic ensures tat tey are conducted in te tougness-dominated regime. In addition to monitoring te injection pressure and pumping rate, te surface displacement associated wit ydraulic fracture growt was measured using an array of linear variable differential transformers (LVDTs): one central and four periperal, wic are sketced in Figure 2.

4 4. EXPERIMENTAL RESULTS 4.1. Pressure and displacement response Wen injection commenced, te pressure increases prior to te beginning of fracture growt. Once te injection pressure reaces about 4 MPa, te pressure increases approximately linearly wit time, wit te slope proportional to te injection rate and inversely proportional to te compliance of te injection system. Eventually crack growt begins, were te pressure required is a function of te strengt of te rock, te applied stresses, te size and sarpness of te initial notc, te injection rate, and te viscosity of te fluid. Te first grap in Figure 3 sows te injection pressure versus time along wit te LVDT-measured displacement on te surface at te nominal fracture center, tat is, directly opposed to te injection tube location. For tis example, at 1600 seconds te injection rate was decreased, accounting for te cange in slope at tis time. Te second grap in Figure 3 sows te surface displacement rate as a function of time. Te point were te injection was stopped (3058 sec) is clearly visible in tis grap. Trougout te initial pressurization te surface displacement increases slowly and nearly linearly. Te surface displacement as an upward inflection, wic is interpreted as te beginning of crack growt, at about 2890 sec (te vertical black line in Figure 3), wile te pressure increases to finally reac a maximum at about 2960 sec. During tis time we infer tat te fracture is indeed growing but wit te injection pressure increasing. In previous researc, te peak pressure exceeding te crack initiation pressure is a penomenon tat as been associated troug modeling and teoretical considerations wit growt of te ydraulic fracture troug te varying near-wellbore stress field [16], te compressibility of te injection system [17], and bot of tese mecanisms combined wit viscous fluid flow [18]. It was previously observed in ydraulic fracturing experiments in rocks by Zao et al. [19], wo also observe an increase of te breakdown pressure wit pressurization rate, and wit decreasing te boreole radius [20]. Wile it can be inferred tat crack initiation significantly precedes te system reacing te peak pressure for te example of test 15M, and indeed, for all tests in medium sized blocks, tis beavior is apparently not present in te smaller blocks. Figure 4 sows te injection pressure, surface displacement at te crack center and te displacement rate for te small block, test 10S. Te time of te first upward inflection in te surface displacement curve is indistinguisable from te time of te peak pressure. Fig. 3. Valve pressure, central LVDT displacement and displacement rate vs experimental time for Test 15M. To explain tis difference between te medium and small blocks, we turn to numerical modelling for a plane strain ydraulic fracture. Reference [18] as sown tat, among oter factors, te difference between te crack initiation and peak pressure increases wen te initial notc dept increases relative to a certain lengt scale tat is smaller wen te injection system compressibility is smaller. In tese experiments, te notces were typically able to be made deeper for te medium blocks (3-5 mm deep) tan for te small blocks (1-1.5 mm deep). Also, examination of te linear portion of te pressure versus time plots indicates tat te injection system was less compressible in te case of te medium blocks. Hence, it is consistent wit previous numerical predictions [16,18] tat te medium size blocks would exibit crack initiation tat clearly precedes te peak pressure wile te small blocks do not.

5 Note tat caracters S and M are related to te tests wit small size and medium size blocks respectively. Experiments 12M, 13M, 14M, and 15M ave approximately te same values of! nom as experiments 11S, 7S, 8S, and 9S respectively. Te sape of te crack surface was determined by direct measurements - eac block was cut troug te boreole in four directions: East, Nort, West, and Sout relative to te blocks orientation as it was placed in te polyaxial compression frame (alf of te block 12M wit saw cut in Nort direction is sown in Figure 5). Ten, tese pats were plotted in dimensionless radius ( R / H ) and dimensionless fracture dept ( / H ) coordinates and were contrasted wit te LEFM experimental crack pats wit te same value of! nom. Data from glass and PMMA (considered as linearly elastic materials) near surface ydraulic fracturing tests described in [8] was taken for tis comparison. Fig. 4. Valve pressure, central LVDT displacement and displacement rate vs experimental time for Test 10S Crack pats analysis Te objective of tese experiments is to compare te crack pats in rock specimens wit tose from omogeneous linear elastic specimens wit te same value of te parameter!. To tis end, te value of te fracture tougness ( K = 2.84 MPa m ) for te largest size semi-circular beam tests (wit diameters equal to 90mm and 30mm tick) was taken to calculate te nominal values of! (! nom ) from Eq. (1). Tese values togeter wit te values of applied confined stress (! r ), distance from te initial notc to te specimen surface ( H ), and injection rates ( Q0 ) are presented in Table 1. Table 1. Values of! r, H, Q0 and! nom for conducted experiments. Exp. ID 5S 7S 8S 9S 10S 11S 12M 13M 14M 15M! r [MPa] H [mm] Q0 [mm 3 / sec]! nom Fig. 5. Block 12M cut in alf (top) and a close-up of te central part (bottom) wit te igligted notc and fracture pats. In te LEFM cases, te crack pats measured in different directions were observed to be igly consistent wit one anoter. Te only appreciable asymmetry in te LEFM experiments came in te form of te initially circular ydraulic fracture eventually favoring propagation in one direction to te detriment of te oters so tat te final plan-view sape of te fracture

6 was egg-saped rater tan circular. Additionally, in te LEFM cases, te fracture reaces te surface ( dayligted ) over only a small portion of its leading edge due to tis late-stage asymmetry [8,9]. Test 8S closely mimicked tis LEFM beavior. Figure 6 depicts te pat measured in four radial directions as well as te curve for te average crack pat for one of te Glass tests from [8], wit te nominal value of! (! = ). One can see tat for Test 8S all four measured crack pats are almost te same in terms of teir sape and lengt; te only significant difference is tat te fracture dayligted in te West direction and, correspondingly, propagated longer in tis direction. Reasonable consistency among te fracture pats in different directions was also observed for test 5S (Figure 7), altoug to a lesser degree tan Test 8S. crack pats from te LEFM experimental crack pat varied significantly. Figure 8. Fracture pats in four directions for test 12M. Figure 6. Fracture pats in four directions for test 8S. Figure 9. Fracture pats in four directions for test 14M. Figure 7. Fracture pats in four directions for test 5S. For te oter tests, some notable differences between fracture pats sapes and lengts in different directions were observed. Figures 8-10 demonstrate tese observations for tests 12M, 14M, and 10S. Typically te corresponding crack pat for te same value of! from te LEFM experiment for eac test was situated between te pats in four directions, even if te deviation of te Figure 10. Fracture pats in four directions for test 10S. Te following procedure was implemented to calculate te experimental values of K for eac test. Figure 11 sows te relation between! and te radius to dept ratio, R / H, corresponding to different values of / H, were is te dept of te fracture surface and H is te initial dept. Hence, te curve / H = 0

7 corresponds to R / H wen te fracture intersects te surface, or dayligts; te curve / H = 0. 2 corresponds to R / H wen te fracture surface is 20 percent of te initial dept away from te free surface, and so fort. So, te goal of te proposed metod is to use curves from Figure 11 to obtain te average value of! given R / H at various values of / H (0.8, 0.6, 0.4, 0.2, and 0). As! satisfies Eq. (1), given te initial dept and te in-situ stress, one may estimate K directly. Figure 12.Te best fit values of! vs te nominal! values for tests 5S,7S-11S,12M-15M. Figure 11. Relation between normalized radius and! at various normalized depts determined by analysis of curving crack surfaces for glass and PMMA. Calculated values of! in four directions (East, Nort, West, and Sout), wic are te aritmetic means of te fitted! at five / H values listed above, are sown in Table 2; te gaps for some tests represent te directions in wic fractures did not propagate far enoug to allow us to estimate te related! values. Values of te stress intensity factor K were calculated using te average (or best fit ) value of! (! ) for eac test. Table 2. Values of! at four different directions for experiments 5S,7S-11S,12M-15M. Exp. ID East Nort West Sout av! av K [ MPa m ] 5S S S S S S M M M M Ten, te best fit values of! versus te nominal values were plotted to observe te correlation between tem (Figure 12). Recall tat te working ypotesis states tat K increases wit te size of te crack, tat is te size of te experiment, and tat tis will be expressed by saucers tat curve more strongly and terefore ave smaller best fit values of! tan te nominal values. And furter, tis tendency is proposed to be stronger for te medium blocks tan te small blocks. In actuality, one can see tat tere is no clear correlation between te nominal and best fit! values. Some experiments fall above! nom =! av line, some below. And it does not depend on te size of te block: bot small blocks and medium blocks tests ave te best fit! values below and above te line. Watever systematic variation tere is in! due to te size effect on te fracture tougness would appear to be masked by oter factors tat all lead to te observed scatter of te experimental results wen viewed in tis way. Wy are te results so scattered? A couple of suggestions can be put forward: (i) Bot te size effect on K tat is under investigation and te crack pat variation (Figures 8-10) tat leads to imprecise estimation of! ave te same origin, tat is, te eterogeneity of te rock. In order to perform experiments at an interesting range of sizes from te perspective of te size effect, tey cannot be too large relative to te size of te eterogeneities. However, at tis scale te small scale fluctuations in stress, preexisting planes of weakness, and variability of material properties among te crystals also cause significant crack pat variability. Tis issue could be a significant detriment to tis approac for investigating size effect if one cannot find a suitable intermediate range of specimen sizes

8 (ii) were te size effect on K is observable but te eterogeneity scale is small enoug to not strongly affect te crack pats. Te initial notc is relatively blunt - around a couple of millimeters in opening. Upon inspection we see tat te fracture can grow from te top of te notc, te bottom of te notc, or in between (e.g. Figure 5 and Figure 10). Tis initial point of crack growt significantly influences te direction of furter fracture propagation and, ence, te final value of measured! in eac direction. It is clear, ten, tat if te size effect on fracture tougness is to be observed and measured using sallow saucer-saped ydraulic fracturing experiments, some additional considerations must be accounted. A wider range of block sizes relative to te size of te eterogeneities in te rock would potentially address issue (i) by firstly giving te possibility for a wider range of variation of!, tus increasing te strengt of te ypotesized systematic response. Secondly, by examining a wider range of block sizes one may find a suitable intermediate size range tat satisfies te simultaneous requirement tat te size effect is observable over te range of sizes but te crack pats are not too noisy due to te influence of te eterogeneities. At least one could more definitively determine if suc an ideal intermediate scale exists. Issue (ii) is, in some ways, more difficult to address as tere are practical limitation on ow sarp and deep one can cut a notc inside a boreole in a laboratory rock specimen. However, in addition to developing tools to make te notc as sarp as possible, increasing te specimen size so tat tis initial variation of initiation dept and direction is smaller relative to te overall size of te experiment would be expected to improve te experimental results. 5. CONCLUSIONS Te experimental study was conducted on near-surface ydraulic fracturing in medium-grained gabbro specimens of two different sizes. Te critical pressures at wic ydraulic fractures initiate and te pats of te propagated fractures were investigated. Te following tings were found: Te effect of te peak pressure exceeding te crack initiation pressure was observed in te tests wit medium sized blocks, but was indistinguisable for small sized blocks. For most of te tests, fracture pats were significantly different in four directions investigated and deviated from LEFM experimental curves. Te absence of clear correlation between nominal values of dimensionless parameter!, governing te fracture curvature, and its experimental best fit values was found, wic was to te detriment of te metod for investigation of te size effect on rock fracture tougness over te range of specimen sizes considered. Te experimental results presented ere suggest tat wider range of block sizes relative to te size of te eterogeneities in te rock as to be investigated. It sould provide one wit a wider range of variation of!, making it possible to find a suitable intermediate size range tat satisfies te requirement tat te size effect is observable over te range of sizes but te crack pats are not too noisy due to te influence of te eterogeneities. ACKNOWLEDGMENTS R. Maknenko and E. Detournay gratefully acknowledge support from te National Science Foundation under Grant No Any opinions, findings, and conclusions or recommendations expressed in tis material are tose of te autors and do not necessarily reflect te views of te National Science Foundation. REFERENCES 1. Hobbs, D.W An assessment of a tecnique for determining te tensile strengt of rock. Brit. J. Appl. Pys. 16: Carpinteri, A Scaling laws and renormalization groups for strengt and tougness of disordered materials. Int.J. Solids Struct. 31(3): Slyapobersky, J., Mo.A. Issa, Ma.A. Issa, MD.S. Islam, J.W. Dudley, Y. Sulkin, and A. Cudnovsky Scale effects on fracture growt resistance in poroelastic materials. In Proceedings of SPE Annual Tecnical Conference and Exibition, New Orleans, September 1998, SPE Borodic, F.M Fractals and fractal scaling in fracture mecanics. Int. J. Fracture 95: Bazant, Z.M Scaling of quasibrittle fracture: asymptotic analysis. Int. J. Fracture 83: Dyskin, A.V Crack growt criteria incorporating non-singular stresses: Size effect in apparent fracture tougness. Int. J. Fracture 83: Bunger, A.P., R.G. Jeffrey, and E. Detournay Evolution and morpology of saucer-saped sills in analog experiments. In Structure and Emplacement of Hig-Level Magmatic Systems, ed. K. Tompson and N. Petford, 302: Bunger, A.P Near-Surface Hydraulic Fracture. PD tesis. University of Minnesota, Minneapolis, MN.

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