A Numerical Hydraulic Fracture Model Using the Extended Finite Element Method

Transcription

1 nernaional Conferene on Mehanial and ndusrial Engineering (CME'2013) Augus 28-29, 2013 Penang (Malaysia) A Numerial Hydrauli Fraure Model Using he Exended Finie Elemen Mehod Ashkan. Mahdavi, and Soheil. Mohammadi Absra The hydrauli fraure phenomenon is numerially invesigaed in his researh. This is somehow differen from he lassial hydrauli frauring, whih is a way of inenional reaion disoninuiy inside a solid; usually oil fields. The mos noieable par of he hydrauli fraure proess is he effe of fluid pressure on rak surfaes whih direly drives raks owards furher propagaion. n his paper, he fous is on he proedure for applying he fluid raion on rak surfaes. The exended finie elemen mehod (XFEM) is employed for numerial modeling of pressurized rak problems. n onras o he lassial finie elemen mehod (FEM), onsidering a rak in a domain is independen from he meshing of geomery of he domain. This mehod allows o model differen rak paerns on a fixed mesh. n his mehod by using he oneps of pariion of uniy, he sandard finie elemen approximaion is enrihed wih addiional enrihmen funions, whih are losely relaed o he orresponding analyial soluion. s advanages failiae he problem of modeling arbirary raks and disoninuiies of he finie elemen mesh. Finally, a numerial example of pressurized rak is presened o illusrae he effiieny of he proposed approah. For verifying he resuls, sress inensiy faors (SFs), whih are derived using he ineraion inegral mehod, are ompared available referene resuls. Keywords Hydrauli fraure, Pressurized rak, Exended Finie Elemen Mehod (XFEM), neraion inegral. A. NTRODUCTON MONG various many numerial mehods whih have been used for modeling he hydrauli fraure phenomenon, he exended finie elemen mehod has no been widely used in his field of sudy. Despie of many advanages, i has no been developed by researhers in his field for sudying he hydrauli frauring problems. Hydrauli frauring is a omplex phenomenon in whih he deformaion of a maerial is aused by he fluid pressure on rak surfaes. Firs models for his phenomenon were proposed in 50 s [1, 2]. Barenbla used he fraure mehanis priniples in hydrauli frauring problems for he firs ime [3, 4]. Green and Sneddon solved he ellipial hole under inernal fluid pressure problem [5] and Spene and Sharp used he sress inensiy faor o assess rak propagaion in hese problems [6]. Ashkan Mahdavi, Shool of Civil Engineering, Universiy of Tehran, Tehran, ran. (Phone: ; ashkan.mahdavi@u.a.ir). Soheil Mohammadi, Shool of Civil Engineering, Universiy of Tehran, Tehran, ran. ( smoham@u.a.ir). The exended finie elemen mehod has been widely used in differen rak and disoninuiy problems. Moamedi and Mohammadi used his mehod for analyzing dynami saionary raks for boh isoropi and orhoropi maerials [7, 8]. Sukumar and Prevos uilized XFEM in quasi-sai rak growh problems [9]. Cohesive rak propagaion was modeled using XFEM by Belyshko and Moes [10]. n his paper, he hydrauli frauring problem is performed in he framework of XFEM, using he ineraion inegral mehod for deriving he sress inensiy faors.. GOVERNNG EQUATONS Consider a body in he sae of equilibrium wih raion and displaemen boundary ondiion, as shown in Fig. 1, Fig. 1 A body in equilibrium sae The srong form of he equilibrium equaion an be wrien as: b. + f = 0 in Ω (1) wih he following boundary ondiions, n = f on n = f on u (2) (3) = u on u (4) Where, and u are he exernal raion, rak surfae and displaemen boundaries, respeively. is he sress ensor and f b is he body fore veor. f and f are he exernal and rak surfae raion veors, respeively. The variaional formulaion of he boundary value problem an be defined as: 92

2 nernaional Conferene on Mehanial and ndusrial Engineering (CME'2013) Augus 28-29, 2013 Penang (Malaysia) W in ex = W (5) b δε. dω = f δud. f δud. f δud. Ω Ω+ + Ω Ω (6) u h ( x) = φ ( xu ) + φ ( xhxa ) ( ) J J J ( ) ( ) l + φk x Fl x bk k K l (7) The rak surfae hydrosai raion is assumed o be perpendiular o he rak surfae, as depied in Fig. 2, g where N is he oal number of nodes, N is he se of nodes whih belong o elemens whih are u by he rak (spli elemens), u is he veor of regular degrees of freedom, aj is he veor of addiional degrees of freedom relaed o he spli elemens, and b l k is he veor of addiional degrees of freedom used for modeling he rak-ips.φ represens he finie elemen shape funion. H( x ) is he Heaviside enrihmen funion, Fig.2 Crak surfae raion + 1 H( x) = 1 ( x x*) e n 0 oherwise (8). EXTENDED FNTE ELEMENT METHOD A. XFEM Approximaion n XFEM, he FEM mesh is generaed regardless of he loaion of any disoninuiy. Then, by using he level se algorihm, he exa loaion of rak or any oher disoninuiy wih respe o all FEM nodes is deermined. Based on he posiion of differen nodes wih respe o he rak loaion, some nodes around he disoninuiy are seleed for enrihmen. n fa, a few degrees of freedom are added o he nodes whih should be enrihed. Suh virual degrees of freedom onribue o he approximaion hrough he Heaviside and rak-ip funions [11]. x* is he neares poin on he rak fae o he poin x, where x is a Gauss inegraion poin, as depied in Fig. 4. Fig. 4 x* is he neares poin on he rak fae o he pin x F ( x) are he rak-ip enrihmen funions as, l θ θ θ θ (, θ) = r os, sin, osθ sin, sinθ sin l { F } 4 l r = 1 (9) r, θ are alulaed in he rak-ip loal oordinae sysem. Fig. 3 Arbirary disoninuiy wihin he FEM mesh Assuming a rak wihin an independen finie elemen mesh, as shown in Fig. 3, he displaemen field for any poin x inside he domain an be wrien as: B. Crak Surfae Traion Applying Proedure The basi idea for applying he rak surfae raion is o assume wo independen sides, upside and downside, for he rak segmen in eah elemen. Then, i an be assumed ha he rak pah is omposed of a se of one-dimensional segmens. Considering an enough number of Gauss poins on boh sides of he rak pah, as shown in Fig. 5, he nodal fore veor for a spli elemen an be alulaed from, 93

3 nernaional Conferene on Mehanial and ndusrial Engineering (CME'2013) Augus 28-29, 2013 Penang (Malaysia) = (10) nodal T f N d where N is he shape funion of he spli elemen and is he raion applied on he rak surfaes. an be shown ha for an elemen whih is u by he rak, exernal nodal fore veor omponens relaed o he regular degrees of freedom vanish. Only he addiional degrees of freedom onribue o he exernal nodal fore of he rak inerfae. The seond erm mus be inegraed over he rak surfae boundary. B. Equivalen Domain negral For uilizing he J inegral onep, a more appropriae form is usually adoped. Aording o he alernaive mehod ha Li e al. [13] proposed, he onour inegral shown in Fig. 6 an be replaed by an equivalen area inegral, as shown in Fig. 7. Fig. 7 Equivalen domain (A*) Fig. 5 Gauss quadraure poins on eah side of The J inegral an now be defined as, V. A. J-negral Conep DERVNG STRESS NTENSTY FACTORS ui q ui J = A * ij W sδ1i da j d x 1 x i x1 (13) where q is a smoohing funion defined as, 1 on q = 0 on 3 1 (14) Fig. 6 Conour around he rak-ip For a ypial raked body, as shown in Fig. 6, he J inegral an be wrien as, u J = W dy d x s (11) Where W s is he srain energy densiy and he body fore and rak surfae raion are negleed. Karlsson and Baklund [12] exended he above equaion o aoun for he effe of rak surfae raion, u u = x x (12) J W s dy d f d The value of q for any inegraion poin an be alulaed by he FEM approximaion proedure. C. neraion negral Mehod This mehod was firs proposed by Sih e al. [14]. They proposed based on his onep ha he boundary value problem an be saisfied by superimposing iliary fields ono he aual fields. The iliary fields are arbirarily hosen and are imposed in order o find a relaionship beween he mixed mode sress inensiy faors and he ineraion inegrals. The onour J inegral in his mehod an be defined as: a J J J M = + + (15) a where J and J are assoiaed wih he aual and iliary saes, respeively. M is defined as, ui u i M q M = A * ij + ij W δ1 j x1 x1 x (16) j 94

4 nernaional Conferene on Mehanial and ndusrial Engineering (CME'2013) Augus 28-29, 2013 Penang (Malaysia) where M 1 W = ( ij εij + ij εij ) (17) 2 The following equaion defines he relaionship of he J inegral and K and K, 2 ( M = K K + K K ) E (18) E where E =, for plane srain problems. 2 1 ν By puing K = 1 and K = 0, he mode sress inensiy faor an be obained. K is obained by seing K = 0 and K = 1. Fig. 9 Normalized sress inensiy faor for differen rak angles (Righ rak-ip) V. NUMERCAL RESULTS An elasi half plane onaining an arbirarily oriened inernal rak near is free boundary is onsidered, as shown in Fig. 8. K (Mode sress inensiy faor) is ompued for differen rak angles. The resuls are ompared wih hose presened by Erdogan and Arin [15]. Fig. 10 Normalized sress inensiy faor for differen rak angles (Lef rak-ip) Sress onours are also presened in Fig. 11, Fig. 12 and Fig.13. Fig. 8 An elasi half plane onaining an arbirarily oriened inernal rak near is free boundary SF resuls for he inlined rak are given for he value of d/a=2, where a and d are he half-lengh and ener-oboundary disane of he rak, respeively. Sress inensiy faors are normalized by 0 a. n his problem, plae dimensions are assumed o be (mm) o saisfy he infinie plae dimensions. The rak lengh is assumed o be 6(mm) and he rak surfae raion is 1000(N/m 2 ). Fig. 9 and Fig.10 show he normalized sress inensiy faors of righ and lef rak-ips, respeively. Fig. 11 yy onour 95

5 nernaional Conferene on Mehanial and ndusrial Engineering (CME'2013) Augus 28-29, 2013 Penang (Malaysia) Fig. 12 Fig. 12 xx xy onour onour The general observaion whih ould be made on he basis of hese resuls is ha for he same rak surfae raions, he resisane of he medium o fraure would be higher for raks nearly perpendiular o he boundary han for hose parallel o he boundary. REFERENCES [1] Hubber, M.K. and D.G. Willis, Mehanis of hydrauli frauring. Journal of Peroleum Tehnology, (6): p [2] Criendon, B.C., The mehanis of design and inerpreaion of hydrauli fraure reamens. Journal of Peroleum Tehnology, 1950: p [3] Barenbla, G.., On erain problems of elasiiy arising in hydrauli fraure sudies. Prikladnaya Maemaika i Mekhanika, (4): p [4] Barenbla, G.., The Mahemaial Theory of Equilibrium Craks in Brile Fraure, p [5] Green, A.E. and.n. Sneddon, The disribuion of sress in he neighborhood of a fla ellipial rak in an elasi solid. Proeedings of he Cambridge Philosophial Soiey, : p [6] Spene, D.A. and P. Sharp, Self-similar soluions for elasohydrodynami aviy flow. Proeedings of he Royal Soiey A, : p [7] Moamedi, D. and S. Mohammadi, Dynami analysis of fixed raks in omposies by he exended finie elemen mehod. Engineering Fraure Mehanis, (17): p [8] Moamedi, D. and S. Mohammadi, Fraure analysis of omposies by ime independen moving-rak orhoropi XFEM. nernaional Journal of Mehanial Sienes, (1): p [9] Sukumar, N. and J.H. Prévos, Modeling quasi-sai rak growh wih he exended finie elemen mehod Par : Compuer implemenaion. nernaional Journal of Solids and Sruures, (26): p [10] Moës, N. and T. Belyshko, Exended finie elemen mehod for ohesive rak growh. Engineering Fraure Mehanis, (7): p [11] Mohammadi, S., Exended finie elemen mehod for fraure amalysis of sruures2008, Oxford: Blakwell Pub. [12] Karlsson, A. and J. Baklund, J-inegral a loaded rak surfaes. nernaional Journal of Fraure, (6): p. R311-R314. [13] Li, F.Z., C.F. Shih, and A. Needleman, A omparison of mehods for alulaing energy release raes. Engineering Fraure Mehanis, (2): p [14] Sih, G.C., P. Paris, and G. rwin, On raks reilinearly anisoropi bodies. nernaional Journal of Fraure Mehanis, (3): p [15] Erdogan, F. and K. Arin, A half plane and a srip wih an arbirarily loaed rak, in Tehnial repor NASA TR-73-8April, 1973, nsiue of Fraure and Solid Mehanis: Lehigh Universiy. V. CONCLUSON n his paper, he exended finie elemen mehod has been uilized for modeling he hydrauli fraure phenomenon. has been shown ha XFEM makes i possible o model pressurized raks wih a saisfaory preision. The aepable agreemen beween his work and available referene daa shows he effiieny of he proposed mehod for modeling hydrauli fraure problems. ACKNOWLEDGMENT The auhors would like o aknowledge he ehnial suppor of he High Performane Compuing Laboraory (HPC Lab.), Shool of Civil Engineering, Engineering Fauly, Universiy of Tehran. 96