THREE-DIMENSIONAL POROELASTIC SIMULATION OF HYDRAULIC AND NATURAL FRACTURES USING THE DISPLACEMENT DISCONTINUITY METHOD

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1 PROCEEDINGS, Thrty-Fourth Workshop o Geothermal Reservor Egeerg Staford Uversty, Staford, Calfora, February 9-11, 9 SGP-TR-187 THREE-DIMENSIONAL POROELASTIC SIMULATION OF HYDRAULIC AND NATURAL FRACTURES USING THE DISPLACEMENT DISCONTINUITY METHOD X. X. Zhou ad A. Ghassem Teas A&M Uversty 3116 TAMU-41E Rchardso Buldg College Stato, TX, USA e-mal: ahmad.ghassem@pe.tamu.edu ABSTRACT A three-dmesoal fully-coupled poroelastc dsplacemet dscotuty method s developed ad used to aalyze the temporal varato of opeg ad slp of a atural fracture a reservor respose to the sudde applcato of flud pressure the fracture surfaces. Numercal results show that a hydraulc fracture opes a creasg maer wth tme as the rock moves towards a draed state uder the appled stress. The appled pore pressure duces a tme-depedet closure caused by the rock dlato. O the other had, poroelastc aalyss of a atural fracture subjected to shear shows that the fracture slp decreases wth the tme respose to a pore pressure-duced crease the ormal stresses o the jot. THEORY OF POROELASTICITY Oe of the ma features of the deformato of fludsaturated porous rock s ts traset ature whch s related to the presece of a flud dffuso process, ad s descrbed by the lear theory of poroelastcty (Bot, Sce the poeerg work of Bot, the theory of poroelastcty has bee reformulated by a umber of vestgators, such as Rce ad Cleary (1976, ad Carroll (198. The coupled costtutve equatos of poroelastc materal uder sothermal codtos are (e.g., Rce & Cleary, 1976: where ε ad j σ j are respectvely the stra ad stress of the sold matr, p ad ζ are respectvely the pore pressure ad pore volume, δ j s the Kroecker delta. The materal costats are the shear modulus G, Bot coeffcet α, the draed ad udraed Posso s ratos v ad v u, Skempto s pore pressure coeffcet B. These equatos descrbe the fudametal aspects of the deformato of flud-saturated porous rock amely; ( the sestvty of the volumetrc respose of the rock to pore pressure chages ad thus the rate of loadg, ad ( varato of pore pressure due to the applcato of a mea stress. The three-dmesoal fled equatos for the poroelastc rock deformato ca be preseted as a Naver equato wth a couplg term, ad a dffuso equato: G G u + u p = 1 ν k, k α, ( ( u ( ν ( (3 ( u ( p u u u p κ GB 1 ν 1 v GB 1 v = ε t 9 v 1 v 3 1 v t (4 where u s the sold dsplacemet the drecto, ε s the volumetrc stra, ad the other otatos are the same as those defed prevously. 1 v α(1 v εj = j kk j jp G σ σ δ + δ 1+ v G(1 + v α(1 v α (1 v (1 + vu ζ = σkk + p G(1 + v G(1 + v( v v u (1 ( DISPLACEMENT DISCONTINUITY METHOD The dsplacemet dscotuty (DD method s a drect boudary elemet method whch s based o the fudametal solutos of a pot DD a fte elastc or poroelastc medum. Ths techque has bee used etesvely mg ad hydraulc fracturg (Crouch ad Starfeld, 1983; Vadamme, 1989; Ghassem ad Roegers, It s a boudary

2 method ad has the advatage of reducg the dmesos of the problem by oe. The formulato of DD ca be based o the soluto of a costat le or square DD a fte elastc medum (Crouch ad Starfeld, Alteratvely, a pot dsplacemet DD ca be tegrated over a area (square, tragle or quadrlateral to form elemets whch form the buldg block of the DD method. The stresses ad dsplacemets due to a three dmesoal pot DD a poroelastc medum s gve by Carvalho ad Curra (1998 ad Cheg ad Detouray (1998. A fracture a poroelastc medum ca be vewed as a surface across whch the sold dsplacemets ad the ormal flud flu are dscotuous. The DDs ad flud sources are the dstrbuted alog the fracture surface such that the superposto of ther effects satsfes the prescrbed boudary codtos at the fracture surface. Gve the DDs ad flud sources, the stresses ad pore pressure at ay pot the reservor rock may be evaluated usg the prcple of superposto: σ ( j d σ ( ', t t' D ( ', t' s σ ( ', t t' D ( ', t' t jk k, t = da( ' dt' + A j j f ( ', ' ( ', ' ( ', ' ( ', ' σ ( d t pj t t Dj t p(, t = da( ' dt ' p A s + p t t Df t ( (5 (6 where D k (or D j ad D f are the dsplacemet dscotuty ad the flud source testy, respectvely; σ, σ, p ad p s are the d jk s j stataeous fudametal solutos,.e., the stresses ad pore pressures due to a ut mpulse of the dsplacemet dscotuty ( d the k-drecto ad a ut mpulse of the flud source testy ( s ; ad σ j ad p are the tal stresses ad pore pressure. Eqs. (5 ad (6 are appled o the fracture surface to obta the soluto system. For plaar fractures, the ormal stresses at the fracture deped oly o the ormal compoet of DD; whle the shear stresses o the fracture deped oly upo the shear compoets of DD. Also, the flud source testy o the fracture cotrbutes oly to the ormal stresses but ot the shear stress o the fracture surface, ad oly the ormal compoets of DD cotrbute to the pore pressures at the fracture surface (z=. Therefore, the tracto compoets at the fracture surface ca be wrtte as: σ ( zz d j d σ ( ', t t' D ( ', t' s σ ( ', t t' D ( ', t' t zzzz zz, t = da( ' dt' + σzz A zz f ( ', ' ( ', ' ( ', ' ( ', ' d t pzz t t Dzz t p(, t = da( ' dt ' p A s + p t t Df t ( ( (7 (8 the ormal drecto, ad σ ( z ( ', t t' D ( ', t' ( ', t t' D ( ', t' d t σ zz z, t = da( ' dt' + σ z A d σ zzy zy σ ( zy ( ', t t' D ( ', t' ( ', t t' D ( ', t' d t σ zyz z, t = da( ' dt' + σ zy A d σ zyzy zy (9 ( ( (1 the shear drectos, where z-as s the ormal drecto of the fracture surface, -as ad y-as are perpedcular ad the fracture surface plae. Eqs. (7-(8 ca be solved together to obta the DDs the ormal drecto ad flud source testes; thereafter, Eqs. (9-(1 are solved to obta the DDs the shear drectos. Oce the DDs ad flud source testes o the fracture surface are determed, Eqs. (7-(1 ca the be used to calculate the stresses ad pore pressure at ay locato the rock matr. I geeral, the tegral equato preseted above caot be solved aalytcally ad therefore, a umercal procedure s requred. I ths work, the fracture s dscretzed to a umber of four-oded quadrlateral elemets the spatal doma, whch makes the tegrals over the whole fracture be replaced by a sum of tegrals over these elemets. The DDs are assumed to be costat over each elemet whch facltates the treatmet of the hypersgular tegratos volved; whle the flud source testes are assumed to vary learly over each elemet. I the tme doma, the DDs ad source testes are assumed to be costat over each tme step, ad the space tegral are performed umercally. The above procedure ca be used to study both hydraulc fractures ad jots. A hydraulc fracture usually s pressurzed ecess of the mmum stu stress ad remas ope durg the loadg process, whch meas both the ormal ad shear stffess of the fracture are zero. O the hydraulc fracture surface, the ormal tractos equal the flud pressures the fracture ad the shear tractos equal zero. However, a dfferet approach must be used for jots, as the jot ormal ad shear stffess are ozero ad the ormal ad shear tractos o a jot chage wth jot ormal ad shear dsplacemets. The procedure for modelg the jot elemet s smlar to that used by Ghassem et al. 7, however, the fracture s dscretzed to a umber of four-odded quadrlateral elemets ths work. Durg the fracture pressurzato, each elemet ca be ether a state of separato, stck or slp ; the elemet s closed the latter two states.

3 I ths paper, we deote the ormal stffess of the fracture as K. The shear stffess of the fracture may be dfferet dfferet shear drectos. However, here we assume the shear stffesses are the same all shear drectos for smplfcato ad deote t as K s. The fracture aperture cremet for ay closed elemet ca be epressed as: d = D + a = + a K σ ' zz dl dl where ' (11 σ s the cremet of the ormal effectve stress ad a dl s the dlato-duced aperture crease due to shear slp. I ths work, for smplfcato, we assume the shear dlato s the same all shear slp drectos. As a result, the shear dlato for ay elemet may be smply calculated by the followg relato: z zy ta adl = D + D φ dl (1 where D ad D z zy are the shear dsplacemet compoets ad y drecto, respectvely ad φ s the fracture dlato agle. dl MODEL VERIFICATION To verfy the umercal model, we compare ts predctos wth the avalable aalytcal solutos for the pey-shaped crack problem. Seddo (1946 solved the problem of a ftely th crack subjected to uform ormal tracto p appled to ts faces. The fracture opeg the ormal drecto s gve by: ( w r ( 41 v pa = 1 ( r a (13 πg where a s the radus of the fracture, r s the radus of the computatoal pot, G s the shear modulus, ad v s the Posso s rato. Also, Seged (195 solved the problem of a ftely th pey-shaped fracture whose faces are subjected to uform shearg tractos, S. The rde of the fracture the drecto of the shear force s gve by: ( u r ( v Sa ( 81 = 1 πg v ( r a (14 where all the otatos are the same as those Eq. (13. The materal s assumed to be lear elastc both Eq. (13 ad (14. However, our umercal model, the materal s poroelastc meag that the fracture aperture would be tme depedet. To compare the two solutos, we cosder the draed poroelastc respose from the umercal soluto, correspodg to a very large tme whe the pore pressures the rock almost completely dsspate (here we use 1 8 s for a rock wth 1-15 m of permeablty. We set the shear modulus ad Posso s rato of the materal to 4 MPa ad.5, respectvely. Fg. 1 shows the fracture mesh whch cotas 8 four-oded quadrlateral elemets ad 841 odes. The sze of typcal elemets s aroud m ad the tme cremet s 1 6 s the computato. Fg. shows the comparsos betwee the umercal ad aalytcal solutos for the opeg of the fracture uder a ut uform ormal tracto. The results for the fracture rde uder a ut uform shear tractos appled at the fracture surface are show Fgure 3. Geerally the umercal results agree well wth the aalytcal results. The error of the umercal results creases ear the fracture tp; ths s caused by the use of costat elemets stead of specal tp elemets. NUMERICAL SIMULATIONS We cosder a horzotal crcular plaar fracture a poroelastc rock. The fracture s subject to a suddely appled, costat flud pressure p = 15 MPa at tme t =. It s assumed that the tal stresses the fled are sotropc ad the vertcal ad horzotal compoets are 3 MPa ad MPa, respectvely. The fracture ormal stffess modulus of the fracture 8 s assumed to be 1 Pa/m. The problem ca be decomposed to two sub-problems correspodg to two types of the loadg (Carter ad Booker, 198: Mode 1, a ormal stress loadg σ = ph ( t ; ad Mode, a pore pressure loadg p = ph ( t, where H( t deotes the Heavsde step fucto. The mesh used for ths part cotas 147 four-oded quadrlateral elemet ad 11 elemet odes. Fg. 4 shows the evoluto of the fracture aperture the mddle of the cetral fracture elemet respose to Mode 1 loadg. Note that the fracture opes wth tme as the pore pressure that s tally geerated the porous rock gradually dsspates. The fracture respose uder Mode s llustrated Fg. 5; the fracture closes progressvely startg from zero to a stablzed value after a log tme. Ths pheomeo

4 s caused by the rock dlato whe the flud leaks-off from the fracture to the reservor matr. Fg. 6 shows the fracture aperture profles for the complete problem (both Modes 1 ad for the udraed ad draed cases. I the udraed case, we let t=1s the umercal smulato so that there s almost o pore pressure dsspato or flud leak-off from the fracture to the rock; whle the draed case, we let t=1 8 s order to allow both of Modes 1 ad traset processes to be complete. Note that the fracture aperture the early tme (udraed case s larger tha that of the large tme (draed respose because of the effect of Mode whch duces a fracture closure. I the followg, we aalyze the opeg ad slp of a plaar fracture that s subjected to a flud pressure whch s less tha the -stu mmum stress. Ths codto ca be epected whe stmulatg geothermal reservors. The fracture surface has a dp agle of 6 o ad ts strke drecto s parallel to the local -as. It s assumed that the fracture s a stu stress of σ v =6.13MPa, σ hm =34.81MPa, σ Hma =5.88MPa, ad p=17.4mpa (Ghassem et al., 7. The oretato of σ Hma s parallel to the fracture strke drecto. Ths stress feld ca be rotated to the local fracture coordate system to obta σ zz =41.1MPa, σ z =MPa, ad σ yz =11.MPa. It s also assumed that the effectve frcto agle ad dlato agle of the fracture are 3 o ad 3 o, respectvely. Both the ormal ad shear stffess of 1 the fracture are assumed to be 1 Pa/m. The other materal propertes used here are show Table 1. I the umercal model, the fracture s dscretzed to a mesh wth 1834 four-oded quadrlateral elemets ad 1919 elemet odes as show Fg. 7. For smplcty, we assume the flud pressure the fracture s costat ad uform ad ts value s 5MPa. Fgs 8-11 show the smulato results for the varatos of fracture aperture, shear slp the - drecto, shear slp the y-drecto, ad the ormal effectve stress o elemet B (see Fg. 7. Fgure 8 llustrates the fracture aperture varato o elemet B wth tme; dcatg a fracture closure. As ca be see Fgures 9 ad 1, the magtudes of the shear slp both the -drecto ad y-drecto decrease wth the passage of the tme. Ths s a terestg result made possble by our aalyss; t ca be eplaed by the crease of the ormal stresses wth tme. The jot closure ad shear stregth are drectly proportoal to the ormal effectve stress at the fracture surface. The ormal effectve stress creases (Fgure 11 respose the matr dlato due to flud leak-off from the fracture to the reservor matr ad costrat dlato. Table 1. Data set used the umercal eample. Parameter Value Shear modulus G (GPa 4. Posso s rato v.5 Flud vscosty µ f (N.s/m.1 Flud dffusvty c f (m /s 1-3 Bot's coeffcet α.95 Flud desty ρ f (kg/m 3 1 Rock desty ρ r (kg/m 3 65 Rock permeablty κ (m 1-16 Fgure 1: Mesh for the crcular fracture eample used to verfy the umercal model. Fgure : Comparsos betwee umercal ad aalytcal results for fracture opeg.

5 Fgure 3: Comparsos betwee umercal ad aalytcal results for fracture slp shear. Fgure 6: Log term ad short term fracture opeg profles respose to combed mode 1 ad loadg; r s dstace to the pot ad a s the radus of the fracture. Elemet B Fgure 4: Varatos of fracture opeg wth tme at elemet A due to Model. Fgure 7: Dscretzato of rregular fracture usg four-oded quadrlateral elemets. Fgure 5: Varatos of fracture opeg wth tme at elemet A due to Mode loadg. Fgure 8: Varato of fracture aperture o elemet B wth tme.

6 Fgure 9: Varato of slp -drecto for elemet B wth tme. Fgure 11: Varato of ormal effectve stress for elemet B wth tme. loadg causes the crease of the fracture closure due to flud leak-off from the fracture to the rock. Ths s cosstet wth prevous studes (Ghassem ad Roegers, 1996; Detouray ad Cheg, Fgure 1: Varato of slp y-drecto for elemet B wth tme. CONCLUSIONS A three-dmesoal dsplacemet dscotuty method has bee developed to aalyze the behavor of the fracture opeg ad slp respose to the applcato of flud pressure the fracture. The feld quattes such as sold dsplacemet, flud flu, stress ad pore pressure, are evaluated by usg the tegral equato method terms of the destes of the dscotuous sold dsplacemet ad flud flu across the fracture. Usg these boudary tegral equatos, we oly eed to dscretze the fracture surface stead of the whole doma. The atural fracture deformato was assumed to be lear elastc ad cosdered dlato related to the fracture slp whch was modeled usg Mohr-Coulomb crtero. The three-dmesoal model was appled to study the poroelastc respose of the fracture opeg ad slp. It was foud that the applcato of the ormal stress loadg causes the crease of the fracture opeg wth the tme because of the dsspato of the pore pressures the rock; whle the pore pressure Ofte, feld applcatos volve jecto pressures that are suffcet to jack the fracture ope. Smulato of ths scearo shows that applcato of the flud pressure a fracture uder crtcal shear stresses causes the fracture to slp ad dlate. Thereafter, the fracture slp decrease as the matr dlato caused by pore pressure dffuso creases the ormal stress o the fracture surface, ad reduces the dlato duced crack opeg. Such a traset slp ca be mafested jecto pressure varatos as well duced reservor sesmcty observed ehaced or egeered geothermal systems. ACKNOWLEDGEMENTS Ths project was supported by the U.S. Departmet of Eergy Offce of Eergy Effcecy ad Reewable Eergy uder Cooperatve Agreemet DE-FG36-6GO95. Ths support does ot costtute a edorsemet by the U.S. Departmet of Eergy of the vews epressed ths publcato. REFERENCES Bot, M.A. (1941. Geeral Theory of Three- Dmesoal Cosoldato. Joural of Appled Physcs. Vol. 1, pp Carter, J.P. ad Booker, J.R. (198. Elastc cosoldato aroud a deep crcular tuel. It. J. Solds Structures. Vol. 18(1, pp Carvalho J. ad Curra J.H. (1998. Threedmesoal dsplacemet dscotuty solutos for

7 flud-saturated porous meda. It. J. Solds Structures. Vol. 35, pp Cheg, A.H.-D. ad Detouray, E. (1998. O sgular tegral equatos ad fudametal solutos of poroelastcty. It. J. Solds Structures. Vol. 35, pp Crouch, S.L. ad Starfeld, A.M. (1983. Boudary elemet methods sold mechacs, Alle Uw, NY. Detouray, E. ad Cheg, A. H.-D. (1993. Fudametals of poroelastcty. I: Comprehesve rock egeerg: prcples, practce ad projects, aalyss ad desg method. Vol. Edtor: C. Farhurst. Pergamo Press, Oford Ghassem, A. ad Roegers, J.-C. (1996. A threedmesoal poroelastc hydraulc fracture smulator usg the dsplacemet dscotuty method, Proc. d North Amerca Rock Mech. Symposum, Motreal, Ca, 1, Ghassem, A., Tarasovs, S. ad Cheg, A. H.-D. (7. A Three-Dmesoal Study of the Effects of thermo-mechacal loads o fracture slp ehaced geothermal reservor, It. J. Rock Mechacs & M Sc., Vol. 44, pp Gordeyev, Y. (1993. Growth of a crack produced by hydraulc fracture a poroelastc medum, Iteratoal Joural of Rock Mechacs ad Mg Sceces & Geomechacs Abstracts, Vol. 3 (3, pp Rce, J.R. ad Cleary, M.P. (1976. Some basc stress dffuso solutos for flud-saturated elastc porous meda wth compressble costtuets. Revews of Geophyscs ad Space Physcs. Vol. 14, pp Vadamme, L., ad Curra, J.H. (1989. A threedmesoal hydraulc fracturg smulator. It. J. Numer. Methods Eg. Vol. 8, pp