Downloaded by guest on September 23, 2021 www.pnas.org/cgi/doi/10.1073/pnas.1818529116 a rolsaewt lsdntrlfractures natural closed Rahimi-Aghdam with Saeed shale oil of or permeability enabling cracks hydraulic of Branching lsdb eodr re o icu o)o hl ne tec- under shale of 10 flow) within viscous stress (or tonic creep 10 secondary about cracks, by average, produced, closed the tectonically on natural, analysis are, the neces- recent which However, that be rate. showed would production 19) gas permeability (18, shale the of the match to of increase sary permeability 10,000-fold at overall gas A spaced the the mass. cracks, increase explain hypoth- natural somehow to universally crack preexisting would necessary been the the has is that reduce it which esized to Consequently, m, a way rate. 0.1 presented only production This about the m. to is 10 spacing branching cca at since spaced dilemma originally cracks, draulic of results. band similar a to by led simulated 17), was (16, shale. crack separations hydraulic interelement element the the Discrete which (12–15). stress or in into clarified and models, been with closing, also leak-off have stability, crack, effect their water the shadow cracks, and parallel along of grains; Interactions controlled crack; proppant advancing of water-filled without of water a of characterization of tip the include flow at They singularity stress (1–11). the stress tectonic under S fracking findings arXiv:1212.11023]. (2018) these spherocylin- al. et demonstrates anisotropic S, law, developed [Rahimi-Aghdam constitutive recently microplane a drical on band crack element based finite if A model, even closed. layers, almost weak initially the are along cracks Biot these cracks hydraulic transverse branching of lateral opening increased engender the greatly to and fracking a the cause with must permeability layers, coefficient, enhanced weak greatly a the with and demon- along forces layers numerically seepage weak is that created It the strated damage. also produced years. microcracking have that million or must events a nanocracking shale tectonic than in the less cracks that in natural considered completely shale is recent been of it a have creep Here and must secondary years, cracks by million How- these closed 100 m. that about 0.1 indicated is about analysis age is average spacing of their whose system ever, preexisting overall cracks, a the natural to permeability increase attributed orthogonal This been would 10,000×. generally which about has m, mass increase shale 0.1 of would about permeability spacing only explain gas crack to the be However, wellhead, to more. the have at or rate m production 10 gas at the spaced casing, well typically with- horizontal are the propagate which of perforations to the cracks from branching current vertical out as predict well It as unavailable. software, mechanics, still commercial fracture is Classical observations here. attempted some is with conflict that mechanics not fracture would associated successful, the of astonishingly formulation and fraccing, consistent developed a (or highly fracking become has aka frac), technology, fracturing hydraulic While Zden by Contributed and 60208; IL Evanston, University, Northwestern Engineering, 87545; Mechanical NM Alamos, Los Laboratory, National a Viswanathan Hari rae det ha iaac)b etnceet ol not could event, tectonic a by cracks, dilatancy) shear shear in to spaces open (due the created However, closing. crack vented hypothesis. the invalidated This deposit). calcite eateto ii n niomna niern,Nrhetr nvriy vntn L60208; IL Evanston, University, Northwestern Engineering, Environmental and Civil of Department hs tde,hwvr rdce obacigo h hy- the of branching no predicted however, studies, These tmgtb betdta ae ntecak ol aepre- have could cracks the in water that objected be might It fpoaaino igehdalccaki lsi rock elastic in crack hydraulic single mechanics a fracture in of made propagation been of have advances ignificant | poromechanics kP Ba P. ek ˇ b or Srinivasan Gowri , 4 | er o10 to years itcoefficient Biot at oebr2,21 sn o eiwOtbr2,21;rvee yHainGoadJh Hutchinson) John and Gao Huajian by reviewed 2018; 29, October review for (sent 2018 21, November zant, ˇ a itTa Chau Viet-Tuan , 6 er i o le ale by earlier filled not (if years 8 | er l,ms aebeen have must old, years c epg forces seepage opttoa hsc iiin o lmsNtoa aoaoy o lms M87545; NM Alamos, Los Laboratory, National Alamos Los Division, Computational X c n Zden and , b yni Lee Hyunjin , | damage kP Ba P. ek ˇ ca. . m, 0.1 a on Nguyen Hoang , zant ˇ e eateto aeil cec,Nrhetr nvriy vntn L60208 IL Evanston, University, Northwestern Science, Materials of Department a,d,e,1 1 ulse ne the under Published interest.y of conflict no declare authors University. The Harvard wrote J.H., performed and and Z.P.B. University; research Brown and the H.G., directed G.S., Reviewers: Z.P.B. H.V., and E.R., S.R.-A. paper. S.K., y and the W.L., data; analyzed H.N., S.R.-A. H.L., V.-T.C., research; mathemat- the S.R.-A., conceived and model; research ical designed Z.P.B. and S.R.-A. contributions: Author em ob eeyitiieadeprcl h xsigcom- predicts existing studies, mechanics The fracture awareness empirical. found as and well is this as intuitive what However, software, merely to companies. mercial be similar some to picture of seems a websites shows nat- the preexisting 1 on along Fig. running fractures. cracks ural branched of necessity the predicted. is branching crack dense and (20) natural het- extensive preexisting an model significant along then mass, previous layers cracks, weak shale the damaged the to if produc- due into that, erogeneity gas introducing, show by observed will enhanced the study is explain This to rate. although occur branching, tion crack must no branching predict the would (20) study previous that branching). crack namely, enhance (20) fact, greatly study could useful shocks preceding a pressure the reveals fluid to nevertheless correction correction triggered unrealistic, vital (this and probably a phase is represents trigger sudden which the a Such on coefficient by response. abruptly pressure Biot dynamic triggered change water This of fact, crack. the change the in increased on sudden was, pressure a water 20 transverse of for ref. coding in unintended program the computer the by predict branching. did crack (20) lateral 20 model about extensive This is shale). water than assumption since compressible cracks, more the the times differ- abandoning in water (another was of crack studies incompressibility of bridged previous the widening between from a bridges ence of on vanishing faces pressure gradually the opposite water by (ii) the the caused and plane, for shale) crack coefficient porous the Biot the into pres- effective water (i) pore of of of account variation gradients diffusion into due Darcy took forces in studies, body sure the previous (i.e., the forces of seepage all with creep trast the for time of plenty left uninhibited. over proceed have to certainly must closing formation, This years rock million years. water-filled 10 million seep nearby about a to a take had would from water it if the surface, and, If ground the immediately. from water in by filled been have owo orsodnesol eadesd mi:[email protected] Email: addressed. be should correspondence whom To urnl tnsa bu %adrrl xed 15%. exceeds rarely and 5% about the which at strata, of stands shale increase currently deep an the various from to cycles, extraction lead gas or optimize of could rate percentage This to its etc. pumping, viscosity, possible of of it history changes the make as such should parameters It bet- allow control. would fracking ter of model realistic a of Development Significance tmyb oe httefakn opne r wr of aware are companies fracking the that noted be may It the from model the gradually, changed is coefficient Biot If indicated branching the that showed however, analysis, Later con- by which, model new a presented (20) paper recent A a exnLi Weixin , NSlicense.y PNAS b at n niomna cecsDvso,LsAlamos Los Division, Sciences Environmental and Earth a aihKarra Satish , b sea Rougier Esteban , NSLts Articles Latest PNAS y d eatetof Department | b f6 of 1 ,

ENGINEERING A is much lower than permeability Kh along these planes (horizon- tal). Therefore, the 3D Darcy law is, in general, anisotropic. In Cartesian coordinates (x, y, z), the resulting volume flux vector q = (qx , qy , qz ) may be written as

q = −µ−1K · ∇ψ, [1]

where ∇ is the vector of gradient operator; K is the 3 × 3 per- meability matrix, which is diagonal if (and only if) the Cartesian axes x, y, z are chosen to be parallel and normal to the bedding planes. Although the natural (or preexisting) cracks in shale strata at 3 km depth must have been closed by 100 million years of creep, B the damage bands along these cracks, which always accompany propagation of fracture process zone, certainly remain (in fact, based on the known surface energy of shale, it can be shown that even empty pores and cracks ca. <15 nm in size, at depth 3 km, cannot close, and this is confirmed by the known size of pores containing shale gas). Permeability Kxx along these bands is surely much higher than it is in the intact shale (although pores of <15 nm contribute nothing globally). To prevent the formation of horizontal cracks, the pump- ing pressure is assumed not to exceed the overburden pressure, which is about 75 MPa. Hydraulic fracturing is considered to pro- duce a system of mutually orthogonal vertical cracks, normal to the directions of the minimum and maximum principal tectonic stresses. The flow of the second type, along the hydraulically cre- C ated cracks, may be assumed to follow the Reynolds equations of classical lubrication theory, which are based on the Poiseuille law for viscous flow. Thus, the horizontal and vertical flow vec- tor components in x, y, z directions along with the cracks may be calculated as

h2 h2 h2 + h2 Q = − y ∇ p, Q = − x ∇ p, Q = − x y ∇ p, x 12 µ x y 12 µ y z 12 µ z [2]

A Fig. 1. Schematic branching due to natural fractures. (A) Water is injected at high pressure through damaged zones and weak layers, (B) crack branch- ing initiates due to the presence of damaged zones and natural fractures, and (C) dense cracking happens in all directions, due to the presence of dam- aged zones, weak layers at closed natural fractures (downward view normal to bedding plane).

no branching. So an intersecting system of open natural frac- tures is assumed to either exist a priori or to develop according to some empirical criteria with no basis in mechanics, supported by some recent experiments indicating the possibility of branching (21–24). A physics-based model for branching, which is our goal, seems lacking. Flow in Porous Solid, Without or With Cracking Damage Two types of flow play a role in hydraulic fracturing: (i) the B flow along the hydraulically created cracks, typically a few mil- limeters wide, and (ii) the flow through nanoscale pores and microcracks or nanocracks in shale with preexisting damage. The latter is negligible after continuous hydraulic cracks form, but here it is found to be crucially important for crack initiation and branching. The volume flow, q, of water through the pores and nanocracks or microcracks of isotropic material may be approxi- mately calculated from the Darcy law: q = −(K /µ)∇ψ, where K is the permeability, µ is the dynamic viscosity, and ψ is the phase potential calculated as ψ = p − γg z. Here p is the pore pressure, γg is the pressure gradient due to gravity, and z is the depth from a datum. However, the permeability Kv in the direction Fig. 2. (A) Fluid flow in intact and damaged shale. (B) Biot coefficient normal to the bedding planes (x, y), i.e., in vertical direction z, increase in transverse direction due to increasing damage (20).

2 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1818529116 Rahimi-Aghdam et al. Downloaded by guest on September 23, 2021 Downloaded by guest on September 23, 2021 and S and phase, solid the shale, aged where read they shale, undamaged For water- apply. the with phases for the medium relations of Biot-type two-phase equilibrium classical a the which as for modeled pores, saturated be may shale Coefficient Biot The and Solid Two-Phase in Equilibrium high MPa). compressible be 70 can more fracking to times during (up pressure 20 water about the and fact, concrete, in than is, It compressible. as Cartesian, been y are has coordinates (28) smeared the The which model for used. for constitutive model (25)], microplane limiter constitutive localization spherocylindrical a a [with from damage calculated cracking rock, the in unloading); (for shale of where preserve to energy adjusted be fracture to to Furthermore, have related would softening property, postpeak material a G as treated be itso rc ad,asmdeult h iiu pos- (l minimum cracks the hydraulic to parallel equal adjacent assumed of bands, spacing sible crack of to widths normal cracking smeared where scniee mae vrtebn o lmn)wdh The width. element) (or to band normal the cracks deformation of over cracking widths which smeared in considered (25–27), model is band crack the is axes plane. to bedding normal cracks cal aiiAha tal. et Rahimi-Aghdam the band. above the element to first transverse coefficient the Biot for (E time elements. (C injection damaged values). vs. initially smallest evolution Stress the (D) marks tion. blue the values; Strain highest ( B) model. Initial (A) 3. Fig. where D point Injection ABC V Biot Coefficient , f 0.2 0.4 0.6 0.8 nefciewyt iuaetehdalccak numerically cracks hydraulic the simulate to way effective An x 0 1 = 3 fsae here shale; of σ 0123 ≡ ij σ p C eut ftopaeF iuain ncs fasnl aaeband. damage single a of case in simulations FE two-phase of Results V ∇ ” z = ijkl stepr pressure, pore the is .Nt ht h aea nrf 0 ae sconsidered is water 20, ref. in as same the that, Note ). x − xx σ damage Initial = stetasesl stoi lsi opinetensor compliance elastic isotropic transversely the is kk , b S ” 0 ij ∂/∂ /3 p Time yy stettlsrs tensor, stress total the is where , stevlmti tesi h oi phase. solid the in stress volumetric the is x ” r aaeprso omlsrisdeto due strains normal of parts damage are . . ,. l h ij x δ x , = ij ;

l = y S S h steKoekrdla saseilcase, special a As delta. Kronecker the is   l ij x V ij r o hne u,i hywr,the were, they if but, changed not are

x xx σ , ” vlto,aogtedmg ad fthe of band, damage the along Evolution, ) = − ij = h x x u owtrijcin(h e ak the marks red (the injection water to due y , xx σ  and S and  b ij el E ij x ij r h pnn itso verti- of widths opening the are , kk 0

Stress (MPa) , − steBo ofceto undam- of coefficient Biot the is 0.4 0.8 1.2 1.6 r h tesadsri tensors strain and stress the are /3 and h y 0 2 y Strain  δ 0123 y ij el r Fg 2) (Fig. are ij stevlmti oa stress, total volumetric the is htaepstoe nothe into positioned are that = b = y 0 x l y p i C directions; σ ” , , ijkl ij i Time yy ,2 3 2, 1, = σ stesrs esrin tensor stress the is , kl , rsuepropaga- Pressure ) l Pressure (x x , 1 l x y ≡ , l r the are y x y , must x G 2 [5] [4] [3] x f ≡ ). rnhn.Aprmcaia nt lmn F)cd o a for code (FE) element finite crack in poromechanical role crucial A a of play studies branching. do previous in they ignored although been fracturing, have hydraulic they However 20), dams. ref. under for or of (except 39) risk (38, frictional the cofferdams long assess in no liquefaction to have has sand engineering force geotechnical seepage in the The considered 37). and been that (36, at zero, stress deformation vertical is to total stress resistance the to effective equal pres- is the soil water point, the the in When point soil. a direction at the upward sure within be an stress in must effective Seepage and the solid. solid reduces the porous in the stresses on by applied balanced are forces seepage The seepage The flow. of direction as the defined forces in body soil are forces the on force faces, seepage between microcrack solid the the of band ligaments onto tips. the a microcrack fluid in the of stresses pore tensile A–A the by section compressive resisted by shows the applied and which microcracks stresses 2B, aligned, mostly Fig. preexisting, in of seen as ically to mal vector of direction in component strain and expansion, direc- volume ative the in coefficient vector β Biot unit the of For shale. tion of natural eter, Here relation, proposed: is empirical data, tensor, test consistent, a match direction. to tensorially as appears load which coefficient following, the Biot The on the increases (35). depends generalizing it and that requires damage show This (29–34) cracking results the Test with 0.7. and 0.2 between face. crack initial stress of of center Evolution (D) strain. high of stresses tectonic to subjected boundaries, and at springs by supported (shale) 4. Fig. β C A (9” h icu rgo ae oigtruhasi moe a imposes soil a through flowing water of drag viscous The nor- nanocracking or microcracking of case special the For typically, While, ” b ν T ij (” y Ems o n-afo h oan (C domain. the of one-half for mesh FE (B) . b 11  min = rsuie iecaki Ddmi fatopaepru solid porous two-phase a of domain 2D a in crack line Pressurized (A) ij 00 0 x ) V eest naae material, undamaged to refers 1/3 1 steieatcdmg tantno,and tensor, strain damage inelastic the is ) −2/3 ieto ny n has one only, direction (but n b (but 0 Strain + .Ti qaincnb nepee graph- interpreted be can equation This ≤1). ϕ β ,where ≤1), ν 0 = i ” hseuto gives equation this , ij h itcefiin fsae a vary can shales of coefficient Biot the .1, (” f s kk ” = B /3) ν −b D σ y = ” Stress, σ (MPa) -60 -50 -40 -30 -20 -10

xx x −2/3 ∇p is analyzed Stress ofthiselement 0 ν V nsldpr nteeeeta the at element the in part solid in i 02468 x ” ν = . j 1 , V ” ” β ν = NSLts Articles Latest PNAS o ij i kk . sa miia param- empirical an is ” steieatcnormal inelastic the is Inial crack /3 xeso fteband the of Extension ) 11 (ϕ b steieatcrel- inelastic the is ν /3 Time ≤ oseepageforce No = b and ν 0 i ν ≤ j b b 1). ϕ ij ν = = | sthe is f6 of 3 b b 0 0 [6] [7] b + + T ij x

ENGINEERING A B Fig. 3 B and C shows how damage and pressure propagate after water injection. For this case, the crack band with high water pressure is seen to propagate straight forward, without branching. Now look at stress variation. Fig. 3D shows the stress evolution within in the solid part of the first element above the initial damaged elements. Obviously, the damage during post- peak softening is captured in a stable manner. Finally, consider how the Biot coefficient and permeability vary in one cracked element (the first above the initial damaged elements). Fig. 3E C D contrasts the evolution of Biot coefficient in the transverse direc- tion with its constancy in the forward direction, which agrees with experimental observations. Do the Seepage Forces Suffice to Induce Crack Branching? It is well known in classical fracture mechanics that pressuriz- ing a crack cannot produce tension along the crack faces, and E thus cannot initiate lateral crack branching (branching is possi- ble only at the tip of a crack propagating at nearly the Raleigh speed). In a preceding study (20), it was surmised, under various simplifications, that the seepage forces (Eq. 7) would suffice to produce tension along the crack face and thus initi- ate lateral crack branching. Let us examine this more rigorously. Consider again a horizontal 2D square domain 2.5 m × 2.5 m, containing one line crack (Fig. 4A). By virtue of symmetry, only a half-domain is simulated (Fig. 4B). The water pressure in the F G line crack is gradually ramped up to reach the maximum of

AB MPa Natural fractures Elasc boundary

HI Shale MPa

Injecon Inial fracked Fig. 5. (A) Schematic line crack with weak layer and spring boundaries. (B) point layer Strain and damage evolution. (C) Stress in solid part of one element in weak layer for a shale with b0 = 0.4 and different weak layer permeabilities. (D) C D Stress in solid part of one element in weak layer for b0 = 0.2 and different weak layer permeabilities. (E) Stress in solid part of one element in weak layer for b0 = 0.4 and different relative weak layer strengths. (F) Schematic of seepage forces. (G–I) Evolution of seepage forces.

two-phase solid automatically takes the seepage forces into account in the form of nodal forces. Two-Phase FE Simulations for a Single Damage Band To clarify the role of nanocracking or microcracking, consider first a horizontal 2D square block of shale of dimensions 1.1 m × EF 1.1 m, supported at the sides by springs approximately equivalent to an infinite medium, as shown in Fig. 3A. Water is injected at the center of the south side at the constant rate of 2 m3/s. The anisotropic spherocylindrical microplane model, with the default parameters of shale given in ref. 28, is used as the constitutive model; lx = ly = 2.1 m. The initial Biot coefficient is b0 = 0.4. The tectonic stresses are Tx = −30 MPa and Ty = −30 MPa. Consider that there is a single preexisting band of nanocracks or microcracks predominantly aligned with axis y, represented by the two red elements in Fig. 3A (which is what remains after a crack was closed by up to a million years of secondary creep, or Fig. 6. FE simulation of hydraulic crack branching in a small domain of viscous flow of rock). These cracks cause the vertical permeabil- shale with several orthogonal weak layers along perfectly closed natu- ity in these two elements to increase ca. 1,000 times compared ral fractures. (A) Ring of elastic elements providing elastic support of the with undamaged shale, while the Biot coefficient increases up to boundaries. (B) FE mesh, preexisting natural weak layers, and fracking water 1 and the initial strength decreases to 10% of intact shale. inlet. (C–F) Evolution of pressure in a shale with weak layers.

4 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1818529116 Rahimi-Aghdam et al. Downloaded by guest on September 23, 2021 Downloaded by guest on September 23, 2021 aiiAha tal. et Rahimi-Aghdam compressive significantly. face of crack magnitude the the regardless the along reduce stress Nevertheless, to happen, vanishing). seen cannot if are (even forces this seepage value that stress tectonic show the results of strength The tensile the shale. attained of and (tensile) positive became phase major a have can ignored. coefficient be 4D Biot cannot Fig. the and branching. effect that no show results part, is solid The there the face. in i.e., stress of crack, exten- evolution direct line the the shows initial in the only propagates increase of crack, sion the the the as neglect well we into as damage damage, First, to crack due law. pressurized coefficient Darcy Biot the of via simulated from is diffusion shale Water MPa. 50 natural closed initially and layers ( C weak fractures. with shale fractures. natural a in preexisting and pressure points, of of injection Evolution system mesh, Orthogonal shows FE zone (A) (B) red pressure). (the boundaries. water layers high weak of or propagation fractures natural the of system larger a with 7. Fig. G C E aea rc rnhn ol apni h tesi h solid the in stress the if happen would branching crack Lateral 65 MPa Injection pressure= A h DF iuaino rcigpoesi oiotldomain horizontal a in process fracking of simulation FE 2D The points Injecon (β D H F i.4C Fig. 0). = Natural fractures B σ xx ln h crack the along , hw htthe that shows –H ) fractures. of alternation the solid. to porous due intact heterogeneity and mass shale, layers intact shale weak and the the coefficient and as Biot layers well weak the as the in between difference permeability the the in is branching crack the for of center the at solid porous the domain. ellipse must in pressurized they tension that an biaxial 5F clear a intuitively on Fig. produce it 5. make acting orientations Fig. Their forces 5 forces. in Fig. portrayed seepage crack. the as the around nodes, schematically mesh the shows on almost acting is tors limestone by cemented are cracks natural irrelevant. the not or of effect smaller the a generally, seen, As strength shale). of 5E relative ratio Fig. three in layer, sidered weak the of and facilitates, branching. shale crack the the and up, Biot layer speeds in weak difference tensile the greater the between a reaches Evidently, coefficient finally layer. weak it the until of values strength increases layer tensile weak to the in negative stress from the shale, the in into plotted diffuses is water crack initial above layer 5 weak Fig. the of element first the Biot of case the than bigger for times 5 layers weak branch- both coefficient shale along tensile and the strains 5B initiate, attains transverse Fig. can stress happen. branch this can crack If ing lateral layer. in weak a crack the the strength, to of parallel element layers stress weak normal one transverse the layer. if consider weak occur Further, can exist. the branching along that and confirms crack This initial the along simultaneously value initial the in the while, shale, deposit), crack limestone original intact by separate (uncemented of contact consist in may faces fracture) rep- is natural that (or fractures layers layer natural weak closed the the within out. resent coefficient diffuses natural Biot water two and transverse pressurized, are The uniformly is there both crack two-phase now in The bands) tions. of that damage domain preexisting except (or 2D layers before, same weak the as consider solid now porous we 5A, Fig. In with Fractures Solid Natural Porous Closed Two-Phase in of Branching Crack years Hydraulic of millions next. it to demonstrate they We due flow. though closed or even creep, completely secondary fractures, been (preexisting) have expla- natural must the the surprisingly, phenomena is Not additional branching? nation lateral what lateral the So, model explain and cracks. could explain hydraulic to of suffice not branching do alone forces seepage i.8. Fig. A rmalteeosrain,i rnprsta ao stimulus major a that transpires it observations, these all From vec- force seepage the of evolution the see to instructive is It strength tensile transverse the of effect the clarify to Finally, ratio of effect computed The the that 18) ref. of update an (as conclude thus must We i.5B Fig. C (A–C and eel httehdalccaktnst propagate to tends crack hydraulic the that reveals b ae rsuepoaainfrn eklyr,n natural no layers, weak no for propagation pressure Water ) D b 0 = 0 o h nta itcoefficients, Biot initial the for 0 = ihpermeability with .4, b ij .4 (here K ausices codn oEq. to according increase values r02(i.5D). (Fig. 0.2 or 5C) (Fig. 0 B o natshale. intact for S G–I rel hw h pedn fhg n lower and high of spreading the shows S rel sams elgbe ec,whether Hence, negligible. almost is lutae h vlto fseepage of evolution the illustrates stedmgdt-natstrength damaged-to-intact the is K weak S rel /K ed osalrsrs,but, stress, smaller to leads K b nat weak 0 NSLts Articles Latest PNAS nthe on 1 = C ln h ekband weak the along S b eas h weak the because , rel 0 . n ..As 0.2. and 0.4 = ausaecon- are values σ xx x and vlto in evolution 6 rmthe from y | direc- f6 of 5

ENGINEERING It is worth mentioning that the expansion of solid due to the Conclusions effect of Biot coefficient has been thought to prevent any tension i) The natural fractures have a major effect on hydraulic frac- parallel to the crack face, and thus cause the closing of any lateral turing and are crucial for its success (although they are crack. The preceding results show that this skeptical view does currently neglected by commercial software). not extend to a heterogeneous shale mass containing weak layers ii) Even though the natural fractures must have been closed alternating with intact shale. by millions of years of creep, or sealed by mineral deposits, Next consider the horizontal section in Fig. 6A, with 4 weak a weak layer of nanocracks and microcracks along these layers; bweak = 1, Srel = 0.1, Kweak /K0 = 1000 (Fig. 6B). Water fractures must be expected to facilitate water diffusion. is injected at one point at constant flow rate. Fig. 6 C–E iii) Poromechanics with Biot coefficient depending on the the propagation of high water pressure. Water enters through damage of the solid phase must be used in fracking analysis. prefractured elements, then diffuses along the weak layers iv) Increase of the Biot coefficient in the transverse direction, and, upon attaining sufficient pressure, the crack branches and caused by oriented cracking damage inflicted by fracking, is Poiseuille flow dominates. The importance of weak layers is thus essential to achieve crack branching. evidenced. v) The typical spacing between natural fractures is roughly To demonstrate the present theory on a larger scale, consider 0.1 m. This matches the hydraulic crack spacing that is a bigger horizontal section of shale, a square domain 5 m × 5 m, necessary to explain the typical gas production rate at the containing a uniform orthogonal system of closed natural frac- wellhead. tures with aligned preexisting weak layers (Fig. 7). To be more vi) The widespread opinion that preexisting natural fractures realistic, unequal tectonic stresses are considered in x and y somehow explain why the overall permeability of shale mass, directions; Tx = 30 MPa, and Ty = 40 MPa. inferred from the gas production rate, appears to be about Water is injected at three points at the bottom of Fig. 7A. 10,000 times higher than what is measured on shale cores Fig. 7 C–H shows the evolution of water pressure. The water flow in the laboratory has been basically correct. However, these and damage strain are seen to follow the path of weak layers. fractures are completely closed and do not convey any gas, Extensive branching occurs. Obviously, this branching can cre- and their role is indirect. ate closely spaced hydraulic cracks and thus increase the overall vii) (i) No porosity ⇒ no branching. (ii) No seepage forces ⇒ no permeability of shale stratum by orders of magnitude, compared branching. (iii) No weak layers ⇒ no branching. (iv) Constant with nonbranching cracks in intact shale. Biot coefficient ⇒ no branching. (Note that, consequently, It has also been checked that omitting the natural fractures hydraulic crack branching in granite is impossible.) leads to no branching. This is evident from the pressure propaga- tion pattern in Fig. 8. This figure also documents the localization ACKNOWLEDGMENTS. Z.P.B. thanks Emmanuel Detournay for valuable dis- instability of parallel crack system (12) (also known as the stress cussions and a stimulating challenge that provoked part of this study. The shadow effect), which causes the crack emanating from the work at Northwestern was funded by Los Alamos National Laboratory (Grants DOE LDRD 20170103DR and OBES DE-AC52-06NA25396) to North- middle injection point not to grow long (the long simultane- western University. Z.P.B. is McCormick Institute Professor and W. P. Murphy ous growth of both remaining cracks is made possible by the Professor of Civil and Mechanical Engineering and Materials Science at proximity of the boundaries). Northwestern University.

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