JUUK1>IAL Ul SL lt i>ietL & TECHNOLOGY * No 79B 2010 PREDICTION OF SAND PRODUCTION IN OIL GAS PRODUCER USING MOHR COULOMB FAILURE CRITERION DU" BAO SU XUAT HIEN CAT TRONG CAc GIENG KHAI TIlAc OAU KHI[.]
JUUK1>IAL Ul- SL.lt.i>ietL & T E C H N O L O G Y * No 79B - 2010 PREDICTION OF SAND PRODUCTION IN OIL-GAS PRODUCER USING MOHR-COULOMB FAILURE CRITERION DU" BAO SU XUAT HIEN CAT TRONG C A c GIENG KHAI T I l A c OAU KHI Sir DUNG TIEU C H U A N P H A HUY MOHR-COULOMB Nguyen The Due Vietnam Petroleum Institute Nguyen The IWich Hanoi University ofScience and Technology ABSTRACT Sand production in oil-gas producer is the existence of an amount of sand together with the fluid of oil-gas reservoir To provide technical supports for controlling sand production, it is necessary to predict the production conditions at which sand production occurs Sanding onset prediction involves simulating the stress state on the sudace of an oil/gas producing cavity (e.g bore hole, pedoration tunnel) and applying appropriate failure criterion to calculate the fluid pressure or pressure gradient at which rock failure occurs The paper presents a model for well bore/pedoration tunnel failure induced sand production The model is based on combination of linear elastic theory and Mohr-Coulomb failure criterion The model has beeb used to calculate minimum bore hole pressure for well bore stability for three cases of diferent stress regimes with data from a Vietnam field The influences of inclination and azimuth of well bore on its stability are also evaluated Finally, the effect of reservoir pressure depletion on 'safe' bore hole pressure is also evaluated TOM TAT Sinh cdt cdc giing khai thdc diu-khi Id sw tin tai ciia mdt Iwgng cdt ldn ddng sdn phim cua md diu Di cd thi cd dwgc nhirng trg giiip ky thudt cho viec kiim sodt sw sinh cdt, cdn thiit phdi dw bdo nhirng diiu kien khai thdc gdy sw sinh cat Dw bdo xuit hien cat ddi hdi viec md hinh trang thai irng suit tren bi mat cdc hie khai thdc diu (giing khoan, Id bin via) vd dp dung cdc tieu chuin phd huy thich hgp di tinh todn dp suit ddng chdy hoac gradient dp suit md tai dd sw phd huy thdnh he dit dd xay Bai bao trinh bay mdt md hinh dw bao diiu kien pha huy he giing khoan vd lo bin via gay nen sw xuit hien cdt khai thac diu Md hlnh dwa tren ca sd ly thuyit dan hdi tuyin tinh vd tieu chuin pha huy Mohr-Coulomb Md htnh dwgc dimg tinh todn dp suit giing tdi thiiu di giir giing dn dinh da dwgc thwc hien cho ba trwdng hgp chi wng suit khdc vdi dir lieu tir mdt md ciia Viet Nam NhCrng anh hwang cua nghieng va gdc phwang vi ciia giing khoan len sw in dinh cua nd dwgc danh gia Cudi cimg, anh hwdng cua sw suy giam dp suit via len dp suit ddy giing 'an toan' cung dwgc ddnh gid perforations in a cased hole If these induced stresses exceed fonnation in situ strength, the formation will fail and sand could be produced together with fluids of reservoir Therefore, sanding prediction needs a knowledge about the mechanisms upon which the rock failure has occurred It is very important to exactly determine what mechanism has caused the problem of formation instability I INTRODUCTION In the last two decades, geomechanical stability has become regular consideration from oil exploration to production [1] The geomechanical instability is usually faced in the drilling with high rig rates in deep water, the drilling in tectonic fields, salt-domes, highpressure high-temperature fields, and the drilling of more horizontal, highly deviated and muhilateral wells ([2]-[4]) An important problem requiring geomechanical stability analysis is related to sand production ([5]-[7]) Production of reservoir fluids at high rates (low bottomhole flowing pressure) cause an increase in the induced tangential stresses concentrated on the face of an open hole or on the walls of Instability of formation around a borehole (or perforation tunnel) IS usually evaluated with a combination of constitutive models and failure criteria ([2], [8], [9]) Constitutive models are a set of equations used to determine the stresses around the hole They range from simple linear elastic models to 135 .lOllRNAL OK S( IEN( E & lECHNOLOGY * No 79B - 2010 sophisticated poro-elasto-plaslic models All ihc consliUilive models have onlv studied thc effect of a few parameters on the hole stability and have ignored ihc rest (|8|-| 11) Actually, there is no constitutive model which can handle all the parameters that affect ihc hole stabilitv There also are various failure criteria which arc used to determine the onset ol' failure in the rocks An-ioiig them, the Mohr-Coulomb criterion is ihe most common failure criterion encountered in geotcchnical engineering Many geotechnical analvsis methods and programs require use of this failure criterion most conveniently described in a coordinate system [x,y,z) where thc r - a x i s is parallel to the hole, ) ' - axis lo bc horizontal, and A:-axis lo bc parallel lo the lowermost radial direction of the hole (sec ligiire 1) 0"v In this studv, stability analvses for sand prediction have been performed bv using a combination ol' linear elastic consliUilive model and Mohr- Coulomb failure criteria, fhe method has been employed lo analy/e wellbore stability for a field of Vietnam with different stress regimes The calculated results show the effect of inclination and azimuth on wellbore stability is strongly dependent on in-situ stress state For the most stable wellbore of each case, the analyses are also carried out for examining the infiuence of reservoir depletion on the potential of sanding The results of study can give useful information for designing development plan of the field Pig I Coordinate wslem fir a hole of well hare or perforation tunnel As can bc seen in Figure a coordinate transformation svslem from (.v.v.-) svstem can (.v f ' r ' ) be obtained to by two operations: I) a rotation ci round r'-axis, and 2) a rotation / around the v -axis The angle / represents the hole inclination and the angle d represents the azimuth angle II DESCRIPTION OF MODEL 2.1 Stresses around well tunnel bore/perforation In petroleum engineering, the holes of vertical/horizontal well bore or perforation tunnel and their adjacent formation are often approximated as thick-walled hollow cvlinder Using this approximation, wc are able to obtain a solution for the near well bore/perforation tunnel stress state and use it in sand production prediction Fig Coordinate transformation Assume that the principal stresses in the virgin formation are: a^ the vertical stress The transformation can be described mathematicallv bv the following direction cosines: CTf^ the largest horizontal stress, andCT;,, the Ke- Kv- ^v.-•T^'^^ cosines of the angles between smallest horizontal stress A coordinate system -v-axis and x' [x',y',z') cr„, y y -'-axes, respectively is oriented so that v' is parallel to Kx- • Ky • ^ "The cosines of the angles between is parallel to (T,, , and z' is parallel to y -axis and x\ a., (i.e 2'-axis is vertical; see Figure 1) The stresses in the vicinity of the deviated hole are 136 y', -'-axes, respecfively JWUKi-v/vi yjr cyy.,c.i^^E & TECHNOLOGY * No 79B - 2010 L - K / between ; axis The cosines of the angles and x' y' where /?„, is borehole pressure, v is Poison's r'-axes ratio and indicate the angular position around thc hole (see I'ignre 1) respectively These cosines are related to As failure is governed by the principal thc stresses rr,, a^, inclination angle / and the azimuth angle a as: L- = C0S/ c o s / K Ke = cos/sino K K- = - s i n / /,- / = sin/ cos a = C0S/7 / = sin/' s'm Cl =0 / = cos/ transforming to the following matrix equation defines planes of principal stress = -sin/7 a, the (.Y, f r ) coordinate system, the fonnation stresses 0 (1) By o"^ a cr, 0 ^/ 0 cr (4) Taking the determinant of the above matrices, the principal stresses are given by the folKnv ing eigenvalue equation: a^, (7;, and (7,, become: (cT,.-a){{a,-aXa.-a)-Tl} (5) cy =/,, cr„ -\-l: cr Bv solving above equations, the principal acting on thc wall of a well bore or perforation tunnel can by computed as, a - - = ^ ' c ^ W + / r , ^ / , +/'r-^v ^.v (2) =iJ ^H+KJ,y(^l,+KJyr(^ c^- = P ^, =-(^^ +^J + -V(^^ -o-J' +44 (6) ^lr=LJ r-^H +K-eKc(^l, +KJ.r.(^^Here the superscript indicate that these are the virgin fonnation stresses Equations (2) represent the stress state in the case of no hole in the formation The stress state will change when a hole exists in the formation For the case of cylindrical hole, it is convenient to present the stresses in cylindrical coordinate [r,6,z) By assuming that there is no displacement along z -axis (plane strain condition), a derivation of the stress solution around cylindrical hole can be found and the stresses at the hole wall are given by the following equations: ^ = A a, = al + CTI - 2(al -a;)cos(2e)-4< The maximum and minimum stresses acting on the wall of well bore or perforation tunnel will be as follows a^ = max (TJ - m a x a,,a,,a ( (7) 2.2 Failure criteria sin(2^)- A, For collapse of well bore or perforation tunnel, the Mohr-Coulomb failure criterion is assumed This is governed by the maximum and the minimum stresses [8] Fig shows the Mohr-Coulomb criterion and a Mohr's circle that touch the failure line The Mohr-Coulomb criterion can be expressed mathematically as follows, a = crl - v ( f c - a : ) c o s ( ^ ) - r ; sin(2^)) r = TQ-h cr tan (Z^ Tg, = -2r°^ sin + 2T°^ cos (8) where, r and a are shear and normal stresses (3) respectively, TQ is the inherent cohesion and cff is the angle of internal fricfion 137 .lOl RNAL OK SCIENC E & TECHNOLOGY * No 79B - 2010 angle and Poisson's ratio, and (c) the borehole trajectory (a/imuth and deviation) Fig -^ Mohr-Ccndomb T — CT Idilure criterion •fhe program can be used to study infliienee of inclination and azimuth on well bore stability in order to optimize well trajectory and predict sanding problem in the future, fhe calculated results are presented here for illustration Ihc case study has been conducted with real data from a sandstone formation in a field in offshore of Vietnam from triaxial test, the sandstone has a cohesion of 178^ psi, a friction angle of 44.2 degree, and a Poison's ratio of 0,15 At a production depth ol' 11 142 ft, the vertical stress is equivalent to the overburden pressure, equal to 10956 psi, the pore pressure is taken at 4836 psi and the Biot's factor is set to 0.7 as suggested by most authors I he analvsis of available Fff/LOTdata suggested that thc minimum horizontal stress equal lo 9036 psi I lowever no information can bc employed to determine the maximum horizontal stress In order to cover potential uncertaintv range, calculations have been perfonned for three ease: in S/hhC The shear and normal stresses can bc calculated as ('>) where (X, and O", are maximum and minimum effective stresses which can be calculated as (10) cr, = cr, - apQ Where, p^ Base case: is pore pressure and a a„ =\.\a„=^ 9940 psi is Blot's coefficient Low stress case: Combining the equations failure condition becomes: above, [CT[ - c r , )-(cr, -i-crjsin^ = 2TQ cos cf) the CTff =CT,^ =9036/'.v/ ligh stress ease: (11) cr„ =l cr,, = 3147/7.V/ According to Equation (6), in thc case of collapse of well bore or perforation tunnel at low hole pressures, a^ will be the maximum principal stress cr^ where fhe stress state is usuallv classified into three in situ stress regimes based on the relative magnitude between the vertical and horizontal field stresses [12] Nonnal or extensional faulting (NF) stress regimes are associated with CT^ > cr,, > G,^ comprcssional reverse or thrust faulting (RF) stress regimes are associated with a„ > cr,, > c r „ , and strike-slip (SS) stress regimes are associated with a^ > O",, > o",, According to the classification, CT, will be the minimum principal stress a^ III CALCULATED RKSLLTS The model described above have been used to write a computer program using FORTRAN programming language This program is able to predicted collapse condition of well bore or perforation tunnel for any combination of in-situ stress state and pore pressure The developed model requires us first to define the following input parameters: (a) the in situ stresses and pore pressure at the depth of the studied formation,, (b) the cohesion, friction the base case and the low stress case are in NF stress regime and the high stress case is in RF stress regime The difference between the base case and the low stress case is that the first is in isotropic horizontal stress state while the 13f JOURNAL OK SCIENCE & TECHNOLOGY * No 79B - 2010 second is in the stress state of horizontal anisotropy depth The minimum borehole pressure at the of interest for different borehole inclinafions (/ ) and directions / azimuths (ci) are shown in Figures 4-6 From the calculated results ol' the base case presented in Figure it is apparent that a vertical borehole is more stable than a horizontal borehole in all directions However, the optimum drilling trajectorv is not necessarily vertical In this case, the lowest borehole pressure that is required to prevent borehole collapse is for a 40"-deviated borehole in a direction parallel to thc minimum in situ stress Cy, 57(»ri 5500 s3()o 51 (ll) 490i I 470( 4500 4?("0 41 or, H Incliiinfion depree /•"/_ '1 s.-uid free produclion X ^ IM 11 If If I i " '11 35fiii i _ ^1 lUl •.•: :• :._ envelope.'1 s.ind free production zS 2.-^'" " 2l '1 N 15f«) i III K.I sand I'm hire zone — ' V- SI l\ l| 4>, /' ã ID' (1 2( i(ô1 >liOll i ll ll ^ ^( II II I SI II :,-Ki Kc"stTVoir 'it'ssui I' Fig Sand free operating (anisotropic base ease) / •'^S|)|1 r (1 ^ " Slid psi envelope ; 2( II K I plot I SI s.iiid failure zone / 111 1(11111 /" S I III Figure shows the sand free operating envelope plot for the low case with isotropic horizontal stress As the reservoir pressure decreases from 4836 psi to 2800 psi, the minimum borehole pressure decreases from 3818 psi to 2800 psi (i.e maximum drawdown II II 11II 111 2iii')n -^riDO 4111.0 -^l lllll Reseivoir Pres.sure, psi Fig Sand free operating (anisotropic high case) 140 envelope plot o^J^J^l^nLJ wi c»^i.^,,^ii & TECHNOLOGY * No 79B - 2010 In summary, the obtained resuhs indicate that vertical boreholes will minimize the potential borehole instability only when the horizontal in situ stress is isotropic Having anisotropic horizontal stress, which is the common case, will divert the optimum well path from the vertical direction In this situation, deviated and horizontal boreholes arc potentially more stable than vertical boreholes Non-vertical boreholes should be drilled in the direction of minimum hori/ontal stress