STABILITY OF BOREHOLES IN A GEOLOGIC MEDIUM INCLUDING THE EFFECTS OF ANISOTROPY *

Transcription

1 Applied Mamatics Mechanics (English Edition, Vol. 20, No. 8, Aug 1999) Published by SU, Shanghai, China Article I.D: ( 1999) STABILITY OF BOREHOLES IN A GEOLOGIC MEDIUM : INCLUDING THE EFFECTS OF ANISOTROPY * Dinesh Gupta, Musharraf Zaman (School Civil Engineering Environmental Science, University Oklahoma, Norman, OK 73019, U S A) (Communicated by Chien Weizang) An analytical formulation is developed to investigate stability a The one-dimensional deep, inclined borehole problem drilled in motion a geologic a medium rigid flying plate subjected under to explosive an internal attack has an analytic pressure solution only a non-hydrostatic when polytropic stress field. index The formulation detonation consists products a three-dimen- equals to three. In general, a numerical sional (3-D) analysis analysis is required. stresses around In this a borehole, paper, however, combined by with utilizing internal pressur- "weak" shock behavior ization reflection borehole shock to in obtain explosive an approximate products, solution applying overall stress small distribu- parameter purterbation method, tion. The an orientation analytic, first-order borehole approximate, in-situ solution stresses is obtained bedding for plane can problem all be flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. arbitrarily related to each or to represent actual field situations. Both tensile Final velocities flying plate obtained agree very well with numerical results by computers. Thus failure shear failure potentials a borehole are investigated. The failure criteria an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic index) for estimation applied assume that velocity when flying least principal plate is established. stress exceeds strength formation in tension, a tensile failure occurs. Shear failure is represented using modi- fied Drucker-Prager failure criterion for anisotropic materials. A parametric study is 1. Introduction carried out to assess effect material anisotropy, bedding plane inclination in-situ stress conditions on borehole stability. Results parametric study indicate Explosive driven flying-plate technique ffmds its important use in study behavior materials under that wellbore intense stability impulsive is significantly loading, shock influenced synsis by a high diamonds, borehole inclination, explosive high welding cladding degree metals. The material method anisotropy, estimation in-situ stress flyor conditions velocity high way formation raising bedding it are questions common plane interest. inclination. Under The assumptions stability a one-dimensional borehole in an elasto-plastic plane detonation medium is also rigid investigated. flying plate, In normal approach order solving to evaluate problem extent motion plastic flyor zone is around to solve a borehole following system effect equations governing anisotropy flow field detonation material on products this plastic behind zone, a mamatical flyor (Fig. I): formulation is devel- oped using ories elasticity plasticity. The borehole is assumed to be vertical, subjected to hydrostatic stresses, drilled in a transversely isotropic geologic mediap +u_~_xp + au um. A parametric study is carried out to investigate effect material anisotropy on plastic behavior geologic medium. Results indicate that stress distribu- tion around a borehole, extent plastic zone, failure pressure are influenced by degree material as anisotropy as value in-situ overburden stresses. It was observed that borehole becomes less stable as degree anisotropy geologic medium increases. Key words : stability borehole; tensile failure; shear failure; failure criteria; stress where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, with trajectory distribution R reflected shock detonation wave D as a boundary trajectory F CLC flyor number: as anor TBI22; boundary. TE21 Both Document are unknown; code: A position R state parameters on it are governed by flow field I central rarefaction wave behind detonation wave D Introduction by initial stage motion flyor also; position F state parameters products The study response a borehole to changing loading conditions is particular interest to * Received date:

2 838 Dinesh Gupta Musharraf Zaman several engineering fields including petroleum engineering, geotechnical engineering mining. Maintaining a stable borehole is primary importance during drilling production oil gas. A knowledge stress distribution around a drilled hole is great significance in such situations as drilling, during production or injection, when calculating fracturing pressures. This stress distribution usually results from non-hydrostatic stress field in a geologic medium (Aadnoy, 1987 [~] ). Borehole instability is a continuing problem within petroleum industry, resulting in substantial yearly expenditures by industry ((hag, 1994 [2] ). As a result, a major concern drilling engineer is to keep borehole wall from falling in or collapsing. In recent past, use highly inclined, far reaching, horizontal boreholes has been increasing steadily. Due to economic considerations, re is a need to better underst failure mechanisms se extended-reach boreholes. A majority previous studies have assumed that a geologic The one-dimensional medium as well problem as distrilsution motion stresses a rigid within flying such plate a medium under explosive are isotropic. attack has Such an assumptions analytic solution have been only found when to be insufficient polytropic in index describing detonation failure in products actual field equals conditions to three. where In general, geologic a numerical media (e. analysis g. rocks) is required. are ten In anisotropic this paper, in however, nature (Aadnoy, by utilizing 1987 hi). "weak" shock behavior reflection shock in explosive products, applying small parameter purterbation The method, ories an analytic, stress measurements first-order approximate in an anisotropic solution medium is obtained are for well documented problem flying in plate literature. driven Lekhnitskii by various high (1981) explosives [3] provided with a polytropic comprehensive indices solution or than stresses but nearly for anisotropic equal to three. elastic Final bodies. velocities Amadei (1983) flying C4] plate followed obtained Lekhnitskii' agree very s well approach with numerical applied results m by to computers. a wide range Thus an rock analytic mechanics formula problems. with two Aadnoy parameters (1987) high Cs] explosive applied se (i.e. detonation solutions velocity to calculate polytropic stress index) for estimation velocity flying plate is established. distribution an inclined borehole drilled in a transversely isotropic geologic medium subjected to a non-hydrostatic stress field. Ong (1994) [z] extended Aadnoy's (1987) Is] model to include nonlinear poroelastic effects. Lekhnitskii 1. Introduction ( 1981 )C3] Amadei (1983) [4] derived general expressions Explosive for driven transformation flying-plate coordinates technique from ffmds one its important axis to anor. use in Aadnoy study (1987) behavior E~] applied materials se expressions under intense to transform impulsive stress loading, coordinate shock synsis geologic medium diamonds, property explosive coordinate welding system to cladding borehole metals. coordinate The method system estimation n transformed flyor velocity borehole coordinate way system raising to it are radial questions one. common A borehole interest. in rocks, in general, fails eir due to exceeding tensile strength or by exceeding Under shear assumptions strength. A one-dimensional geologic material plane usually detonation behaves differently rigid flying in plate, tension normal than in approach solving problem motion flyor is to solve following system equations shear as a result, separate failure criteria are required to describe each type failure. Bradley governing flow field detonation products behind flyor (Fig. I): (1979) [6] developed a numerical model to calculate conditions borehole tensile failure in vertical deviated wells under normal tectonic in-situ stress conditions. The basic assumption in analysis were isotropic ap +u_~_xp medium, + linear au elastic solid condition plane strain along axis borehole. Bradley's au work au showed 1 that inclination borehole is a very important factor neglecting it would result in erroneous prediction fracture initiation pressure. Ong (1994) [2] conducted a parametric as study to as investigate tensile failure criterion. The results showed that effect anisotropy becomes significant only in boreholes having inclination greater than 45 ~ drilled in a geologic medium with high degree anisotropy. For a low borehole where inclination, p, p, S, u are difference pressure, between density, isotropic specific entropy anisotropic particle solutions velocity is small detonation products use respectively, isotropie solution with may trajectory be justified. R Shear reflected failure shock initiatiod detonation pressure may wave be D defined as a boundary as external trajectory pressure required F flyor to cause as anor collapse boundary. Both borehole. are unknown; Aadnoy (1987) position E~] used R Mohr-Coulomb state parameters failure on criterion are governed to predict by shear flow failure field initiation I central pressure. rarefaction If wave horizontal behind in-situ detonation stresses wave are D by initial stage motion flyor also; position F state parameters products different, a stable borehole can be drilled in direction least in-situ stress. The deformation pattern was investigated by increasing borehole pressure. Hsiao (1988) [7] studied failure horizontal boreholes. The Drucker-Prager's failure criterion was used to predict stability

3 Stability Boreholes in a Geologic Medium 839 wellbore in an isotropic medium. Critical borehole pressure was found to be significantly affected by in-situ tectonic stresses, borehole orientation geologic material properties. The permissible borehole operating range (borehole stable region), determined from oretical failure model developed, provided a convenient technique for control borehole pressure which was equivalent to control borehole fluid density during drilling or production. A number researchers (e. g., Westergaard, 1940Cs]; Gnirk, ]; Risnes et al., 1982[1~ I4_siao, 1988 [7]) have conftrmed existence a plastic zone around a borehole. Westergaard (1940) [8] concluded that at great depths, a plastic zone exists around borehole, which helps to relieve stresses. In his work, Westergaard used concept effective stresses. A detailed discussion on influence pore presssure resulting from a fluid contained in a porous rock was given by Biot (1941) In]. He proposed a general ory three-dimensional (3- D) consolidation, The one-dimensional taking problem into account motion possibility a rigid a flying flowing plate pore under fluid. explosive Gnirk attack (1972) has C9] an realized analytic solution existence only a when plastic zone polytropic around index borehole. detonation The rock products was assumed equals to be to situated three. In in general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock a hydrostatic stress field to obey Mohr-Coulomb criterion plastic yield. The wellbore behavior reflection shock in explosive products, applying small parameter purterbation pressure method, required to an prevent analytic, plastic first-order yielding approximate an uncased solution wellbore is obtained was evaluated for problem assuming that flying no plate fluid driven flow would by various be involved. high explosives with polytropic indices or than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus an 1 analytic Scope formula with two parameters high explosive (i.e. detonation velocity polytropic index) for estimation velocity flying plate is established. The main objective present study is to develop a mamatical formulation to realistically predict fracture initiation pressure shear failure initiation pressure for deeper, 1. Introduction inclined boreholes in an anisotropic geologic medium. The formulation for stress distribution around borehole Explosive is driven based flying-plate on concept technique generalized ffmds its important plane strain use in linear study elastic behavior solid. It is materials assumed under that intense stress impulsive state does loading, not change shock along synsis z-axis. diamonds, The formulation explosive consists welding a three- cladding dimensional metals. analysis The method stresses around estimation a borehole flyor due velocity to internal way wellbore raising pressure it are questions non hydrostatic common interest. stress tensor. The borehole, in-situ stresses formation bedding plane can Under assumptions one-dimensional plane detonation rigid flying plate, normal take any arbitrary orientation in formation, allowing one to simulate any given field conditions. approach solving problem motion flyor is to solve following system equations In elasto-plastic analysis, a formulation is developed using ories elasticity governing flow field detonation products behind flyor (Fig. I): plasticity. The stress distribution around borehole due to plastic yielding formation extent plastic zone around borehole are determined based on formulation. ap +u_~_xp + au 2 Mamatical Formulation The geometry problem considered in formulation is shown in Fig. 1. The derivation follows closely earlier works Lekhrtitskii as as (1981) E3J, Amadei (1983) [4], Aadnoy (1987) Ell Ong (1994-) [2]. Due to drilling borehole, in-situ stresses are significantly modified near borehole wall. This change in stress state is important since large stress deviation can lead to failure in formation around borehole. where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, The basic with assumptions trajectory made R in reflected derivation shock are as detonation follows: wave D as a boundary trajectory (1) "I]:ansversely F flyor as isotropic anor geologic boundary. medium; Both are unknown; position R state parameters (li) on Linear it are governed elastic material; by flow field I central rarefaction wave behind detonation wave D by initial stage motion flyor also; position F state parameters products (iii) Plane strain condition (for calculation stress distribution around borehole) ; (iv) Homogeneous continuous medium. Figure 1 shows a plan view a circular borehole situated in a non-hydrostatic stress field

4 840 Dinesh Gupta Musharraf Zaman where a is radius borehole b is outer radius field. The stress components in a continuous body (that is in equilibrium) must satisfy following equilibrium equations (Lekhnitskii, 1981 [31) : 0 ar x.~ 0 rxz Ox '+ 8y +-~-'z = 0, (la) 0 try ~ 0 rx: Ox + 0y + Tz = 0, (lb) 0 rx~.~ 0 o': a--'--x- + 3y +-~"z = 0, (le) where a,, a r a.. are normal stresses in x, y The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic z directions, solution respectively, only when r,r, polytropic r= index ry, detonation products equals to three. In Fig. 1 Schematic general, are shear a numerical stresses on analysis xy, is required. xz yz In planes, this paper, however, by utilizing "weak" shock problem analyzed behavior respectively. reflection shock in explosive products, applying small parameter purterbation Using method, Hooke's an analytic, law, first-order constitutive approximate equation solution relating is obtained stress for strain problem for a general flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. anisotropic case, in borehole coordinate system, can be written as: Final velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters {e}= high [a]{a}, explosive (i.e. detonation velocity polytropic (2) index) where for [ A estimation ] is termed as velocity compliance flying matrix, plate { is e established. } vector strain components { a I is vector stress components. The strain components are independent 1. Introduction z because generalized plane strain concept adopted. Applying this concept, equations compatibility for strain reduce to form (Aadnoy, Explosive driven flying-plate technique ffmds its important use in study behavior 1987 [1]) : materials under intense impulsive loading, shock synsis diamonds, explosive welding cladding metals. The method estimation 0'2r 02% flyor velocity 02Y~r way raising it are questions (3a) common interest. 0y2 + ~;2 = 0X0y' Under assumptions one-dimensional plane detonation rigid flying plate, normal approach solving problem motion 3),,, _ flyor 07~ is = to 0, solve following system equations (3b) Oy Ox governing flow field detonation products behind flyor (Fig. I): 02r 02r 02~: _ 0, (3~ Oy 2 - Ox 2 - OxOy ap +u_~_xp + au where cx, er e: are strains in x, y z directions, respectively, 7~r, 7= )'y: are shear strains on xy, xz yz au planes, au respectively. 1 The equations equilibrium (Eqs. 1 a, l b l c) can be satisfied by introducing two stress functions F(x, y) ~b(x, as y) such as that (Amadei, 1983 [41) 32F O2F 02F 0_~ ~-~ (4a e) Oy 2 ' Ox 2 ' 8xOy Oy Ox a x -- (~y -- Txy =-, rx: =, ry: =-. where Appling p, p, S, u are generalized pressure, density, plane strain specific condition entropy frbm particle expression velocity for detonation e~ in Eq. products (2), respectively, with trajectory R reflected shock detonation wave D as a boundary sixth component a. can be expressed as follows: trajectory F flyor as anor boundary. Both are unknown; position R state parameters on it are governed by flow field I central rarefaction wave behind detonation wave D by initial stage 0". = motion -!(a310" x + a320"y + a34rrz + a35rx, + a36~bxr), (5) a33 flyor also; position F state parameters products where ax, a r a. are normal stresses in x, y z direction r,r, rxz rrz are shear stresses in xy, xz yz planes, respectively, a31, "'", a36 are coefficients

5 Stability Boreholes in a Geologic Medium 841 compliance matrix [A]. Substituting Eqs. (4a - e) (5) in constitutive relations (Eq. 2) n inserting se results in conditions compatibility (Eq. 3), one obtains following differential equations which stress functions must satisfy: L4F + L~b = 0, L3F + Lzr = 0, (6a) (6b) where Lz, L3 L4 are differential operators second, third fourth orders which are given as follows: L2 = f44 0x 2-2fl45 flx0y + f55 0y2, The one-dimensional 03 problem motion 93 a rigid flying 03 plate under explosive 03 attack has an analytic L3 solution = - P24-0,? only - + when (f25 + f46) polytropic 0,,20y index - (fl. + detonation f56) 0-~ products + f15 09,3. equals to three. (7b) In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior reflection L4 = fl22 0x4. shock in 2fl26 Ox30y. explosive + (2fl12. products, + fl66). applying small parameter pur- Ox20y2-2fill OxOy 3 + Hill 0y4. (7c) terbation method, an analytic, first-order approximate solution is obtained for problem flying plate flo' is driven called by various tensor high reduced explosives strain with coefficient polytropic indices is given or by than following but nearly expression equal to three. : Final velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two flij parameters = aij- ai3013 high explosive ( i,j = (i.e. 1,2,4,5,6). detonation velocity polytropic (8) index) for estimation velocity flying a33 plate is established. Solving two Eqs. (6a) (6b) simultaneously in terms stress function F, following sixth order differential equation 1. is Introduction obtained: Explosive driven flying-plate technique (L4L2 - ffmds ~)F its = important O. use in study behavior (9) materials The under algebraic intense equation impulsive that loading, corresponds shock to synsis Eq. (9) can diamonds, be written explosive as follows welding (Amadei, cladding 1983 [4] ) : metals. The method estimation flyor velocity way raising it are questions common interest. Under assumptions one-dimensional I4(,u)I2(,u) plane - I2(,u) detonation = 0, rigid flying plate, normal (10) approach where solving problem motion flyor is to solve following system equations governing flow field 14(/1) detonation = fill, u4 - products 2fl16, u3 + behind (2fl12 + f66)[/2 flyor (Fig. -- 2f126,u I): + /~22, (lla) 13(, u) = f15[ A3 - (f14 + f56)[ A2 + (/~25 + /~46)fi t -- f24, (lib) ap +u_~_xp + au [2(, u) = fl55, u2-2f45,u + fl44. (llc) There are six roots,,uk(k = 1 to 6), y characteristic =0, equation (Eq. 10). Let/11, /1 s,u 3 be those roots,u~, /.l 2 as,u 3 as ir respective conjugates. Three complex numbers 2 1, are defined as follows (Lekhnitskii, 1981 [31) : I3(/11 ) 13(/12) 13 (,u3) [2(/.tl), 22 =- 12(d.t2), 23 =- i4(/.t3). (12a,b,c) where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, Lekhnitskii with (1981 trajectory [3] ) showed R that reflected general shock expressions detonation for wave stress D as a functions boundary F trajectory can be written F as flyor follows: as anor boundary. Both are unknown; position R state parameters on it are governed by flow field I central rarefaction wave behind detonation wave D by initial stage motion F = 2Re{Fl(z flyor also; i) + F2(z2) position + F3(z3)}, F state parameters products (13a) r = 2Re 2iF'l(.1) + 2~F;(.,) '7-- ' (7a) (13b)

6 842 Dinesh Gupta Musharraf ganum where (i) Re is notation for real part complex expressions in brackets; (li) F(zk) (k = 1, 2, 3) are analytic functions complex variable zk = x +/.tky (x, y) are coordinates point within body where stress, strain displacement components must be determined; (ill) A prime indicates a derivative with respect to complex variable z k. 2.1 Components stress Lekhnitskii (1981 [3] ) introduced three new analytic functions ~k complex variable zk which can be given as follows: ~I(Zl) = F'l(Zl), ~2(g2) = F2(z2), ~3(r-3) = F3(z3). (14a,b,c) The Using one-dimensional Eqs.(4a ~ e), problem (13a, b) motion (14a, a b, rigid c) flying general plate under expressions explosive for attack stress has an components analytic solution can be obtained only when as follows polytropic (Aadnoy, index 1987 [1]): detonation products equals to three. In general, a numerical a~,h analysis = 2Re{/z2#'l(zl) is required. + In /Zz 2~' this 2(z2) paper, + a however, 3[.t23 ~'3 (g3) by } utilizing, "weak" shock (15a) behavior reflection shock in explosive products, applying small parameter purterbation method, cry. an h analytic, = 2Re(~',(zl) first-order + approximate ~2(z2) + 23~;(z3)}, solution is obtained for problem flying (lsb) plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities rxy,h flying = plate -- 2Re{t/l~'l(Z.1) obtained agree + very [.d2~'2(g well 2) with +,~3/-t3~'3(z3)}, numerical results by computers. (lse) Thus an analytic formula rx..h with = 2Re{Z,/~,#~(z,) two parameters + high A2/~2~;(z2) explosive (i.e. + /~,#;(z3)}, detonation velocity polytropic (lsa) index) for estimation velocity flying plate is established. r,~.h = L 2Re{X,#q(z~) + 22~~ + ~;(z3)}, (15e) 1( 1. Introduction a:,h = - a31a,, h + a32trr, h + a34rr:,h + a35rxs + a36rxr) (15f) t~33 where Explosive a,. h, at. driven h, a,. flying-plate h, rxr. h, r,r. technique h rr, ffmds ' h are its stress important components use in induced study by behavior following: materials 1) drilling, under intense impulsive loading, shock synsis diamonds, explosive welding cladding metals. The method estimation flyor velocity way raising it are questions 2) boundary stress acting along borehole wall. common interest. Under It remains assumptions in above equations one-dimensional to solve plane for detonation analytic function rigid flying ~k. The plate, solution normal was approach found here by solving considering problem boundary motion conditions flyor for is to a solve circular borehole following subjected system to equations external governing stresses on flow cylindrical field detonation surface. products behind flyor (Fig. I): The boundary conditions along surface borehole wall can be written as follows : _a dy dx ap +u_~_xp + au ds + r'r ~ss = ~' (16a) _ % ay dx ds + ar ~ss = ~r' (16b) as as - r,~ + rr" ~ss = 8'' (16c) where ~,, gr, 8~ are components in x, y, z directions external stresses acting on where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, cylindrical with surface. trajectory By substituting R reflected Eqs. (4a, shock 4b, 4c, detonation 4d wave 4e) into D as Eqs. a boundary (16a, 16b trajectory 16c), F boundary flyor as conditions anor boundary. can be rewritten Both are as follows unknown; (Amadei, position ): R state parameters on it are governed by flow field I central rarefaction wave behind detonation wave 3 {OFIdy + 3 (~F)dx D by initial stage motion flyor also; position = F - ~,, state parameters products (17a) ~-x/ds + 5~x( = ~r, (17b)

7 Stability Boreholes in a Geologic Medium 843 8_~dy 8_~dx Oy ds + ax d"-~- = - ~'' (17c) By integrating Eqs. (17a), (17b), (17c) with respect to arc length s from a certain point (s = 0) from Eqs. (13a) (13b), one obtains following expressions: 0"--Y -= 0 OFoy - 2Re{41(zl) + 42(Z2) + a3[-t343(z3)} = fso~rds, (18b) Ii,d, ~b = 2Re{,}tl41(gl) + a242(2',2) + 43(2:3)} = - (18e) Combining internal external boundary conditions applying superposition The one-dimensional problem motion a rigid flying plate under explosive attack has an principle, analytic expressions solution only for when total boundary polytropic conditions index are detonation obtained products as follows equals : to three. In general, a numerical ~ = (ax.o analysis - Pw)cOs0 is required. + rxr.osin0 In this paper, - i(a+,o however, - Pw)sin0 by utilizing + ir+r.oeos0, "weak" shock (19a) behavior reflection shock in explosive products, applying small parameter purterbation method, ~:r = an (crr,o analytic, - P,)sin0 first-order + r+r.ocos0 approximate - i(ar, solution o - P,,)eos0 is obtained + for ir+r,osin0, problem flying (19b) plate driven by various high explosives with polytropic indices or than but nearly equal to three. ~:~ = qz.oeos0 + rrz,osin0 - ir... sin0 + irr~,oeos0. (19e) Final velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic The equation formula with two contour parameters a circular high borehole explosive in (i.e. parametric detonation form velocity is given as polytropic follows: index) for estimation velocity flying plate is established. x = a cos0, y = a sin0. (20a,b) Using zk = a (cos0 + tzk sin0) Eqs. (17a), (17b) (17c), one can write total 1. Introduction expression for boundary condition as follows: Explosive driven flying-plate technique ffmds its important use in study behavior materials under intense 2Re{/.t141(Zl) impulsive + loading, /-t242('~,2) shock + /].3/.t343(Z',3)} synsis diamonds, = -- ~xds = ad, (21a) 0 explosive welding cladding metals. The method estimation flyor velocity way raising it are questions common interest. I' 2R~{4~(.1) + 42(.2) +,~343(z3)} = e,d, = ~E, (21b) Under assumptions one-dimensional plane detonation 0 rigid flying plate, normal approach solving problem motion flyor is to solve I' following system equations 2Re{21~l(zl) + /'~242(/" 2) + ~3(Z3)} =- ~ds = af. (21c) governing flow field detonation products behind flyor (Fig. 0 I): In order to determine total stress distribution at borehole wall, on needs expressions for analytic function 4k. This can be achieved by solving above three equations (Eqs. ap +u_~_xp + au 21a, 21b 21c) simultaneously. The expressions for analytic functions ~ are obtained as follows (Aadnoy, 1987 C1]) : tz as as 41 = ~'-~[0(~2,~3 -- 1) + E(/z 2 - ~2,~.3~3) + F(/.z 3 - /A2)~3], (22a) 42 = ~-'~[D(1-2123) + E(A123/.z3 - /.tl) + F(/.z I - /.t3)~3], (22b) where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, with 43 = 2"G[D(~,I trajectory a R - R2) reflected + E(/~I shock "~2~-- ~221) detonation + F(~2 wave - D /~l)], as a boundary (22e) trajectory where F flyor as anor boundary. Both are unknown; position R state parameters on it are governed by flow field I central rarefaction wave behind detonation wave D by initial stage D = (P,, motion - a,.o)sin0 flyor + also; r~r.oeos0 position + i(p,, F - a,.o)cos0 state - parameters ir~,osin0, products (23a) E = (Pw - ar,o)cos0 + r,r,osin0 - i(p,, - ay.o)sin0 + irxr,oeos0, (23b) F = - r... sin0 + ry,.oeos0 - ir... cos0 - irr:,osin0, (23e) I' f,

8 844 Dinesh Gupta Musharraf Zaman G = (//2 - /21) + 3`23`3(/21 - /23) + 3`13`3(/23 - /22). (23d) The next step would be to take derivative analytic function ~, with respect to dimensionless complex variable zk. Performing necessary operation, derivatives analytic function are obtained as follows: 1 ~'1 = ~';-1[D'(3`23`3-1) + E'(/22-3,23,3/23) + r'(/23 - /22)3`3], (24a) ~'2 = 2-~21D'( 1-3`13`3) + E'(3`13`3/23 - /21) + F'(/21 - /23)23], (24b) - ~'3 =.,1-JL-r[ D' (3`1-3`2) + E'(/213`2 - zt, 3 /223`1) + F'(/22 /21)], (24c) where The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic solution only when polytropic index detonation products equals to three. In D' = (Pw - a,.o)coso - general, a numerical analysis is required. Gr.osin0 In this - paper, i(p~ however, - a~.o)sin0 by utilizing - ir~r.ocos0, "weak" shock (25a) behavior E' reflection = - (P, shock - ar.o)sin0 + explosive r,r.ocos0 products, - i(p~ - ay.o)eos0 applying - ir~r.osin0, small parameter (25b) purterbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven by F' various = - r... high cos0 explosives - rr,.osin0 with polytropic + ir~,.osin0 indices - iry,.ocos0, or than but nearly equal to three. (25c) Final velocities flying plate obtained agree very well with numerical results by computers. Thus G'~ = [/2kcos0- sin0][/22 - /21 + 3`23`3(/21 - /223) + 3`i3`3(/23 - /22)] an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic index) for estimation velocity flying plate is established. (k = 1,2,3). (25d) 2.2 Stresses around wellbore The final stress distribution around 1. Introduction borehole wall can be considered as tensor sum following three stress distributions: Explosive driven flying-plate technique ffmds its important use in study behavior materials ( i under ) The intense far-field impulsive stress tensor loading, before shock drilling; synsis diamonds, explosive welding cladding ( ii ) metals. The stress The tensor method induced estimation by drilling flyor velocity borehole; way raising it are questions common ( iii ) interest. The stress tensor induced by boundary stress acting along borehole wall. Under By superimposing assumptions above three one-dimensional conditions, plane final detonation expressions rigid for stress flying distribution plate, normal around approach solving problem motion flyor is to solve following system equations borehole wall can be written as follows: governing flow field detonation products behind flyor (Fig. I): a, = G.o + a~,.h = a,.o + 2Re{/22~ ', + /22~2 + 3`3/22~3}, a, = %,o + ar,h = ap a,,o +u_~_xp + 2Re{~ + au + #'2 + A3#;}, (26a) (26b) = + = - 2Re{/2,r +/22+'2 + 3`3raG}, r,, = r... + r,,,h = as r,,,,, + as 2Re{2,/2,#', + 3`2/22#2 + /23#;}, ry, = rr, o + r~,,.h = p =p(p, rr,.~ - s), 2Re{A,+~ + A2~'2 + #'3}, (26e) (26d) (26e) where p, p, S, u CY.. are = pressure, ~:,o + G:,h density, = G:,o- specific entropy particle velocity detonation products respectively, with trajectory R reflected shock detonation wave D as a boundary trajectory F flyor as 1...1_{ anor a310"x,h boundary. + a320"y, Both h + are a34rr~,h unknown; + a35g:,h position + a36rxy,h}, R state (26f) parameters on it are governed a33 by flow field I central rarefaction wave behind detonation wave D by initial stage motion flyor also; position F state parameters products where a... at. o G. o are normal in-situ stresses in x, y z directions, respectively, Gy, o, r... rye, o are in-situ shear stresses in xy, xz yz planes, respectively. The mamatical formulation derived above allows one to take into consideration any

9 Stability Boreholes in a Geologic Medium 845 arbitrary orientation borehole, in-situ stresses, bedding inclination. In present study, this formulation is used to evaluate stress distribution around a borehole. The calculated stress distribution is n substituted into different failure criteria to determine stability boreholes. The effect different parameters such as material anisotropy, borehole inclination, bedding plane inclination, stress inclination on stability borehole is illustrated using above formulation. Two computer programs were developed, based on above formulation, to study sensitivity different parameters affecting stability boreholes. 3 Coordinate Transformation In order to develop a general scheme that can take into consideration an arbitrary orientation an The anisotropic one-dimensional geologic problem medium, in-situ motion stresses a rigid borehole flying plate inclination under explosive, it is necessary attack has to an establish analytic solution expressions only for when polytropic transformation index detonation products equals gb to three. In general, coordinates a numerical from one analysis axis is to required. anor. In For this a paper, given however, by utilizing "weak" shock behavior reflection shock in explosive products, applying small parameter puranisotropic geologic medium, in-situ stresses terbation method, an analytic, first-order approximate solution is obtained for problem flying plate borehole driven inclination, by various high explosives following with coordinate polytropic system indices or than but nearly equal to three. Final can be velocities defined with flying reference plate obtained to global agree or very arbitrary well with numerical results by computers. Thus an coordinate analytic formula system xyz with (Fig. two 2) parameters (Ong, ]): high explosive (i.e. detonation velocity J polytropic index) 1) for estimation x, y, z represent velocity global flying plate or arbitrary is established. coordinate system; 2) Xr, y,, Z, represent geologic 1. Introduction medium g8 property coordinate system defined by angles a r fl~; Explosive driven flying-plate technique ffmds its important use in study behavior 3) x,, y,, z, represent in-situ stress coordinate materials under intense impulsive loading, shock synsis diamonds, explosive welding cladding system defined metals. by The angles method a, r, estimation ; flyor velocity way raising it are questions common 4) xb, interest. Yb, zb represent coordinate system attached Under to assumptions borehole which one-dimensional is defined by angles plane a detonation b rigid flying plate, normal approach fib. solving problem motion flyor is to solve following system equations governing All flow parameters field detonation need to be products transferred behind to flyor (Fig. I): borehole coordinate system n to radial coordinate system. After carrying out appropriate ap +u_~_xp + au transformation, compliance matrix au au anisotropic 1 material in borehole coordinate system (xb, Yb, zb), in matrix form, can be written as as follows: as [At] = [P,][M~]T[A][M~][P,] T, (27) where [P,] is strain transformation matrix for where converting p, p, S, borehole u are pressure, coordinate density, system specific to entropy global particle velocity detonation products respectively, coordinate system. with [M trajectory o] is R stress reflected transformation shock detonation wave D as a boundary Geologic medium property trajectory matrix for F converting flyor as geologic anor medium boundary. property Both coordinate are unknown; position R state parameters on it are governed by flow field I central rarefaction wave behind detonation wave system to global coordinate system. D by initial stage motion flyor also; position F Fig. 2 state Coordinate parameters reference products The matrix stress components {a} b in frames borehole coordinate system, after transformation can be x Borcholc Zr r

10 846 Dinesh Gupta Musha_,xaf Zaman written as {0"}b = [O,][&]r{0"},, (28) where [ R, ] [ O, ] are strain stress transformation matrix, respectively, for converting stress coordinate system to global coordinate system. { 0" }, is matrix stress components in stress coordinate system. After transformation, complias~ce matrix in radial coordinate system becomes: JAr]r0, = [ T~,][P,][M,]r[A][M,][P,]T[ T,] v, (29) where [ To] is transformation matrix for converting borehole coordinate system to radial coordinate system. 4 Failure Criteria The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic A failure solution criterion only is when a relationship polytropic between index stresses, detonation representing products a equals limit beyond to three. which In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock failure occurs. In wellbore stability analyses, one is generally concerned only with tensile /or behavior reflection shock in explosive products, applying small parameter purterbation shear failures. method, an analytic, first-order approximate solution is obtained for problem flying plate driven Having by determined various high explosives stresses at with points polytropic around indices borehole or wall, than it but is nearly necessary equal to to compare three. Final computed velocities stresses flying against plate obtained formation agree strength. very well At with points numerical where results stress by computers. state exceeds Thus an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic formation strength (eir in tension or in shear) failure is considered to have initiated. Most index) for estimation velocity flying plate is established. strength criteria are expressed terms principal stresses. The stresses at wellbore wall, computed in preceding sections, are converted into three principal stresses (0"1, a2 0"3)- 1. Introduction These principal stresses can be calculated, in borehole coordinate system, using following expressions Explosive : driven flying-plate technique ffmds its important use in study behavior materials under intense impulsive loading, shock synsis diamonds, explosive welding 0"1 "~ 0" r "~ P w, (30a) cladding metals. The method estimation flyor velocity way raising it are questions common interest. 0"o + 0", ~/(0"0-0"~)2 Under assumptions O2 one-dimensional plane detonation 2 + r2z' rigid flying plate, normal (30b) approach solving problem motion flyor is to solve following system equations governing flow field detonation 0"o products + 0"~ behind /(0"o - 0",)2 flyor (Fig. I): 0"3-2 - ~/ 2 + r~:, (30c) where al, 0"2, 0"3 are major, intermediate minor ap +u_~_xp + au principal stresses, respectively. 4.1 Tensile failure criterion In general, borehole tensile failure is predicted by minimum normal stress ory (Bradly, 1979 E6] ; Yew Li, 1988[12] as ; Ong, as 1994 [2]). Assume that geologic medium has a tensile failure stress, o"t, that fracture initiates at inner surface borehole wall when least principal stress (0"3) at surface reaches or exceeds this value i. e., 0"3 ~>- at. (31) where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, The location with failure trajectory on R borehole reflected shock wall detonation fracture wave initiation D as a boundary pressure can be trajectory obtained by F using flyor numerical as anor methods. boundary. Both are unknown; position R state para- 4.2 meters Shear on it are failure governed criterion by flow field I central rarefaction wave behind detonation wave D by initial stage motion flyor also; position F state parameters products When state stress, at a given point a section in borehole wall satisfies a specific shear failure criterion, failure is said to have occurred. Shear failure results in collapse borehole material.

11 Stability Boreholes in a Geologic Medium 847 There are numerous criteria proposed to define failure geologic medium in shear. But it should be realized that re is no single ory which is satisfactory for all materials. In this paper modified Drucker-Prager anisotropic shear failure criterion (Chen, 1992 [13] ) is used for prediction collapse pressure. Aadnoy ( 1987 C1] ) used Mohr-Coulomb Failure criterion for collapse pressure prediction. This Mohr-Coulomb failure criterion neglects effect intermediate principal stresses also, it does not take into account real anisotropic properties geologic medium Drucker-Prager anisotropic shear failure criterion To account for effect all principal stresses, Drucker Prager (1952 [14] ) proposed a failure criterion for elastic, linear isotropic case, by using invariants stress tensor. Faruque Chang (1986 E15] ) extended Drucker-Prager' s ory to include nonlinear inelastic The one-dimensional stress-deformation problem response motion geologic a media. rigid flying Najjar plate (1990 under [16] explosive ) subsequently attack has an Chen analytic (1992 solution [13]) modified only when Faruque polytropic Chang (1986 index ElS] ) detonation failure criterion products to take equals into to account three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock anisotropic behavior geologic media. behavior reflection shock in explosive products, applying small parameter purterbation The method, basic assumption an analytic, first-order this modified approximate criterion is solution that is failure obtained surface for problem geologic medium flying plate can driven be expressed by various in high following explosives form: with polytropic indices or than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus Ff =?2 _ (aj~a + C2)g(O,Yl) = 0, (32) an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic index) where for ~, estimation Jl, JlA C are velocity non-dimensional flying plate quantities is established. given as follows: r Jl JIA ~ ro? = ~--~; Jl 1. = ~--~, Introduction }la -- p, = ~-~, (33) Pa being Explosive atmospheric driven flying-plate pressure technique ro being ffmds its cohesive important strength use in study geologic behavior medium, a materials k are under material intense constants. impulsive loading, shock synsis diamonds, explosive welding cladding metals. The method estimation flyor velocity way raising it are questions Denoting x-axis as 1-axis, y-axis as 2-axis z-axis as 3-axis, stress tensor [ G art 0.z common interest. r~, rr~ Under r~r} can assumptions be written as {all one-dimensional 0"22 0"33 O" "12}. plane detonation rigid flying plate, normal approach Using solving above definition problem stress motion tensor, flyor variable is to solve J1A is given following by following system expression: equations governing flow field detonation JIA products = Cllali behind + C220"22 + flyor C330"33, (Fig. I): (34) where coefficient c11, c22 c33 are anisotropic coefficients. The variable JlA may be correctly perceived as a variable similar ap to +u_~_xp first + invariant au Jl that has undergone modification by anisotropic coefficients Cll, c22 au c33. au To 1 simplify above equation, it is generally assumed that c H is equal to 1 for a transversely isotropic material c22 = c33. The function g(0, Jl ) is defined as as follows: as g(~,j1) = [cos( -~-arccos(- 1 Zcos30))] -2, (35) where where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, with trajectory R reflected A = exp(-/ shock 7Jl), detonation wave D as a boundary (36a) trajectory F flyor as anor boundary. Both are unknown; position R state parameters on it are governed by flow field 1 r 3"r ] 0 = I ~-arccos/--t---~2 central rarefaction wave /, behind detonation (36b) wave D by initial stage motion flyor also; position t 2(J2D) F J state parameters products where 7 is a material constant. r = 2~-~2o, (36e)

12 848 Dinesh Gupta Musharraf Zarnan J~ is first invariant stress tensor, J2o J3o are second third invariants deviatoric stress tensor, respectively. The stress invariants J1, J2o J3o are given as follows (in indicial notation) : Jl = 0-u = au + 0"22 + 0"33, 1 J2D = -~ Si~S.il, 1 J3D = '~ SijSjkSki, (37a) (37b) (37e) where So. represents deviatoric stress tensor given by (in indieial notation) 1 S0' = 0-~i- ya~8~i, (38) The one-dimensional problem motion a rigid flying plate under explosive attack has an where analytic c~ii is solution Kronecker' only s delta. when polytropic index detonation products equals to three. In general, The a major numerical advantage analysis is required. proposed In criterion this paper, is that however, it is a non-dimensional by utilizing "weak" quantity shock is behavior automatically invariant. reflection Being shock an in invariant, explosive it is products, valid for all applying coordinate small system. parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying plate 5 driven Discussion by various high Results explosives with polytropic indices or than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic The mamatical formula with formulation two parameters developed high in explosive preceding (i.e. detonation sections is velocity rar versatile polytropic since it index) can be for used estimation to model any velocity borehole inclination flying plate under is established. any tectonic conditions. Parameters such as types material anisotropies (isotropic transversely isotropic), degree anisotropy, borehole inclination, bedding plane inclination 1. in-situ Introduction stress conditions ir subsequent effects on borehole stability are studied based on proposed formulation. Two computer programs were Explosive driven flying-plate technique ffmds its important use in study behavior developed based on formulation, one for fracture initiation pressure or for collapse materials under intense impulsive loading, shock synsis diamonds, explosive welding pressure. cladding metals. The method estimation flyor velocity way raising it are questions common Table interest. 1 gives input data used for tensile failure analysis Table 2 presents input data used Under for shear failure assumptions analysis one-dimensional (Lama Vutukuri, plane detonation 1978 [17] ). Given rigid flying this plate, information, normal approach program calculates solving stresses problem around motion borehole flyor n is to compare solve m following with system strength equations governing geologic media. flow When field detonation failure products criteria is behind satisfied, flyor borehole (Fig. I): is said to have failed. The information generated in parametric studies can be used in hydraulic fracture design, drilling fluid program, to help in recognizing possible causes ap +u_~_xp + au borehole failures etc. Table 1 System input data for au tensile au failure 1 analysis ( 1 psi = x 103Pa ) 1. Geometry as as ab = 45 ~ /3b = Variable, a r = 0", fir = 0", a, = 0 ~ /~, = 0 ~ where p, 2. p, Properties S, u are pressure, geologic density, medium specific entropy particle velocity detonation products respectively, with n = 0.25,0.35,0.50,0.75, trajectory R reflected Isotropie; shock detonation wave D as a boundary trajectory F E flyor = 4.75 as x anor 106 psi, boundary. u = 0.14, Both v, = are 0.21, unknown; o t = - I 000 position psi R state parameters on it are governed by flow field I central rarefaction wave behind detonation wave D by initial stage motion flyor also; position F state parameters products 3. In-Situ stresses #, = psi, oh,~ = 7000 psi, ah~. = 5000 psi

13 Stability Boreholes in a Geologic Medium 849 Table 2 System input data for shear failure analysis( 1 psi = x 103Pa ) 1. Geometry a b = 45 ~, fl~ = Variable, a, = 0 ~ r, = 0 ~ a, = 0 ~ ft, = 0 ~ 2. Properties geologic medium n = 0.25,0.35,0.50,0.75, Isotropic; E = 4.75 x 106 psi, v = 0.14, u, = In-Situ stresses a v = psi, O'hm ~ = psi, O'hmin = 9000 psi 5.1 The Tensile one-dimensional failure in problem inclined boreholes motion a rigid flying plate under explosive attack has an analytic The borehole solution is only rotated when 90 o from polytropic vertical index direction detonation (/3 b = 0 products ~ to equals horizontal to three. direction In general, (/3 b = a 90 numerical o), analysis changes is required. in fracture In initiation this paper, pressure however, is calculated by utilizing for different "weak" cases shock behavior reflection shock in explosive products, applying small parameter purterbation material method, anisotropy, an analytic, formation first-order bedding approximate orientation solution in-situ is stress obtained conditions. for problem flying plate driven Effect by various degree high explosives anisotropy with polytropic indices or than but nearly equal to three. Final velocities Degree anisotropy, flying plate n obtained is defined agree as very ratio well between with numerical modulus results elasticity by computers. in Thus plane an normal analytic to formula plane with isotropy two parameters (E,) high modulus explosive elasticity (i.e. detonation in plane velocity isotropy polytropic ( E~ = index) Er = for E), estimation or: velocity flying plate is established. n = E:/E. (39) 1. Introduction Degree anisotropy is used as an indicator so that changes in material properties can be readily Explosive studied. driven Here, flying-plate note that higher technique degree ffmds its anisotropy, important use lower in study n values. behavior materials Figure under 3 shows intense impulsive fracture loading, initiation shock pressure synsis as a function diamonds, borehole explosive inclination. welding A total cladding 5 curves metals. are shown The method ranging from estimation n = 0.25 flyor (highly velocity anisotropic way material) raising it to are n questions = 1.00 (isotropic common interest. material). Results indicate that up to 35 degree borehole inclination, isotropic Under assumptions one-dimensional plane detonation rigid flying plate, normal anisotropic solutions are almost identical but for borehole inclination greater than 35 degrees, approach solving problem motion flyor is to solve following system equations difference becomes subtle. This difference becomes maximum when well is at 90 degree governing flow field detonation products behind flyor (Fig. I): inclination, i. e. in horizontal plane. The difference between fracture initiation pressure among different degrees anisotropy becomes significant for borehole inclination above 45 degrees. For n = 0.25 re is a drastic reduction ap in +u_~_xp pressure + au at 65 degree borehole inclination a situation uncontrollable hydraulic fracturing au may be au experienced. 1 This uncontrollable fracture initiation at se borehole positions is caused by state stress at borehole wall where minor principal stress (0-3 ) is always tensile as greater as than tensile strength (o- t ) geologic medium. Such uncontrollable fracture initiation can create a hazardous drilling condition should be avoided as far as possible Effect horizontal stress ratio where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, Figure 4 with shows trajectory effect R horizontal reflected stress shock ratio detonation M fracture wave D initiation as a boundary pressure as a trajectory function F borehole flyor as inclination. anor boundary. The ratio Both M is are defined unknown; as follows: position R state parameters on it are governed by flow field M = I O'hmin/O'hraJax. central rarefaction wave behind detonation wave (40) D by initial stage motion flyor also; position F state parameters products Here, maximum in-situ principal stress is assumed to be overburden stress by this assumption, stress condition av > O'h max ~ O'h mln i5 considered to exist. It can be observed, from Fig. 4, that for degree anisotropy (n = 0,75), difference

14 850 Dinesh Gupta MusharrafZaman between isotropic anisotropic solution becomes quite significant for high borehole inclination. For M = 1.00, this difference is small but for M = 0.50, difference is large for borehole inclination above 50 degrees, uncontrollable fracture situation may be experienced The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic solution ~ 0.4 only when polytropic index detonation products equals to three. In o..= general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock v -n = behavior reflection shock in explosive products, applying small parameter puro on =0.50 I terbation method, = an 0.2 analytic, first-order approximate solution o is obtained on =0.35 for problem flying plate driven by various = =n =0.25 r, high explosives with polytropic indices or than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic index) for estimation velocity flying plate is established. Borehole inclination fib/(~ (1 psi = x 103pa,1 ft = cm) 1. Introduction Fig. 3 Effect degree anisotropy (Tensile failure) Explosive driven flying-plate technique ffmds its important use in study behavior materials under intense impulsive loading, shock synsis diamonds, explosive welding cladding metals. 2.0 The method estimation flyor velocity way raising it are questions n = O. 75 v 7M = O. common interest. 5u[ Under assumptions one-dimensional plane detonation 0 OM =0.00 ~ :M=I. rigid flying plate, normal approach solving problem motion flyor is to solve following system equations governing flow field 1.5 detonation products behind flyor (Fig. I): -.~ l.o "~ 0.5 ap +u_~_xp + au -"2. C ~ :3 -- = -=-~J~ -I~.=~.._~._~ as ; :. "T +T ~ ~. as where p, p, S, u are pressure, density, specific entropy particle velocity detonation products i I I respectively, with trajectory R reflected shock detonation wave D as a boundary 20 4O O trajectory F flyor as anor boundary. Both are unknown; position R state para- Borehole inclination ~h/(~ meters on it are governed by flow field I central rarefaction wave behind detonation wave D by initial stage motion ( 1 psi flyor = also; X position 103Pa,! ft = F em) state parameters products Fig. 4 Effect horizontal stress ratio (Tensile failure)

15 Stability Boreholes in a Geologic Medium 851 For M = 0.50 higher borehole inclination, fracture initiation pressure decreases as compared to isotopic ease. It is observed that smaller value stress ratio, smaller fracturing pressure will be. One can conclude from above observations that degree anisotropy becomes significant for borehole inclination beyond 40 degrees. An accurate knowledge about horizontal stress ratio M becomes important for a low stress ratio (M = 0.50). Physically this means that as deviation between two horizontal stress increases, stability borehole decreases which is furr compounded by high borehole inclination high degree anisotropy Effect overburden-horizontal stress ratio The overburden-horizontal stress ratio V is defined as folllows: The one-dimensional problem V motion = o'v/o" h a max" rigid flying plate under explosive attack (41) has an analytic In tectonically solution active only when areas, it is polytropic not uncommon index to record, detonation overburden products stress equals that to is three. less than In general, horizontal a numerical stresses. analysis In such is required. tectonic settings, In this paper, one can however, encounter by utilizing a stress condition "weak" in shock which behavior reflection shock in explosive products, applying small parameter pur- O'h max ~ O'g > O'h rain" terbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven The effect by various overburden-horizontal high explosives with stress polytropic ratio (V) indices on fracture or than initiation but nearly pressure equal is to shown three. in Final Fig. velocities 5. It can be flying observed plate that obtained difference agree very between well with isotropic numerical results anisotropic by computers. solutions Thus is an negligible analytic formula up to 4 with 5 degree two parameters inclinations. high It becomes explosive significant (i.e. detonation only velocity at very high polytropic borehole index) for estimation velocity flying plate is established. inclinations (beyond 75 degrees). It can be seen from above analysis that degree anisotropy is an important parameter 1. Introduction only for highly inclined boreholes low stress ratios. It can be concluded that neglecting effect differences in vertical horizontal stress magnitudes can result in less conservative Explosive driven flying-plate technique ffmds its important use in study behavior materials values under fracture intense initiation impulsive pressure. loading, shock synsis diamonds, explosive welding cladding metals. The method estimation flyor velocity way raising it are questions common interest. 2.0 vv = 0.50 n = 0.75 Under assumptions o one-dimensional QV=0.75 plane detonation rigid flying plate, normal approach solving problem ~----ev = motion 1.00] flyor is to solve following system equations governing flow ~ field 1.5 detonation products behind flyor (Fig. I): p e~ 9.~ 1.o.~. 0.5 where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, with 0 trajectory R reflected shock detonation wave D as a boundary trajectory F flyor as 0 anor boundary. 20 Both 40 are unknown; 60 position 80 R 90 state parameters on it are governed by flow field Borehole I central inclination rarefaction fib/q) wave behind detonation wave D by initial stage motion ( 1 psi flyor = also; x position 10SPa,1 ft = F cm) state parameters products Fig. S ap +u_~_xp + au as as Effect overburden-horizontal stress ratio

16 852 Dinesh Gupta Musharraf gaman Effect in-situ stress orientation Figure 6 shows effect inclination overburden in-situ stress on fracture initiation pressure as a function borehole inclination. All curves in Fig. 6 correspond to stress condition where av > ah max > ah m~n a, = 45 ~ 9 The degree anisotropy is taken as n = 0.75, isotropic case. 1.5 : v : /~. = 90" 'v /3,=60* n = 0.75 n n /~, = 30" o o /3,=0* The one-dimensional a, problem motion a rigid flying plate under explosive attack has an analytic solution 01I 1.0 only when polytropic index detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior reflection shock in explosive products, applying small parameter purterbation method, :_- - an analytic, first-order approximate solution is obtained for problem flying plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic index) for estimation velocity flying plate is established Introduction Borehole inclination/~/(*) Explosive driven flying-plate ( 1 psi technique = ffmds x 103Pa,1 its important ft = use cm) in study behavior materials under intense impulsive loading, shock synsis diamonds, explosive welding cladding metals. The method Fig. estimation 6 Effect flyor in-situ velocity stress orientation way raising it are questions common interest. Under It is observed assumptions that for low one-dimensional borehole inclination plane detonation lower inclination rigid flying stresses plate, (t3, normal = 0 0, approach 30 ~ borehole solving pressure problem increases up motion to a certain flyor limit is to solve for high following borehole system inclination equations it starts governing decreasing. But flow for field high detonation overburden products stress inclination, behind flyor t3, = (Fig. 90 ~ I): pressure keep on increasing even at high borehole inclination. It should be noted that re is small effect anisotropy on this particular trend. As material becomes more anisotropic magnitude pressure increases as ap +u_~_xp + au compared to isotropic case. The difference between isotropic anisotropic solutions is less for vertical borehole, it increases as borehole is rotated y downward =0, becomes maximum when borehole is in horizontal plane. as as In summary it can be concluded that in-situ stress conditions affect stability Of borehole significantly. Knowledge about in-situ stresses becomes more important when degree anisotropy borehole inclination both are high. where p, p, Effect S, u are pressure, formation density, bedding specific plane entropy orientation particle velocity detonation products respectively, Figure 7 with show curves trajectory for R 0 ~ 30 reflected ~ 60 shock ~ bedding detonation plane inclinations wave D as a (/3r) boundary isotropic trajectory case. The F formation flyor as azimuthal anor boundary. angle a, is Both assumed are unknown; as 30 ~ position R state parameters on it are governed by flow field I central rarefaction wave behind detonation wave It can be seen from Fig. 7 that difference between isotropic anisotropic solutions D by initial stage motion flyor also; position F state parameters products increases with increasing bedding inclination. The relative positions curves indicate that fracture initiation pressure decreases with increasing borehole inclinations higher bedding plane inclination, more gradual decrease in fracture initiation pressure will be.

17 Stability Boreholes in a Geologic Medium ~ 7 ~ 7 : :- ~. n= lsotropic 0 ~ #,=30 ~ o o ~=0 ~ The one-dimensional 0.2 problem motion a rigid flying plate under explosive attack has an analytic solution only when polytropic index detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior reflection 0 shock in explosive products, applying small parameter purterbation method, an analytic, 0 first-order 20 approximate 40 solution 60 is obtained for 80 problem 90 flying plate driven by various high explosives with Borehole polytropic inclination indices fib/(~ or than but nearly equal to three. Final velocities flying plate obtained ( 1 psi agree = very x well 103Pa,1 with ft numerical = cm) results by computers. Thus an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic index) for estimation velocity Fig. 7 flying Effect plate formation is established. bedding plane orientation ( Tensile failure) 1. Introduction Physically this suggests that formations with high bedding inclinations are more stable against tensile Explosive fracturing. driven Note flying-plate that separation technique between ffmds its important individual use curve in is largest study at high behavior borehole materials inclinations, under suggesting intense impulsive that effects loading, shock bedding synsis plane inclinations diamonds, are much explosive amplified welding at se cladding borehole positions. metals. The method estimation flyor velocity way raising it are questions common It can interest. be concluded from above observations that for a high degree anisotropy (n = Under assumptions one-dimensional plane detonation rigid flying plate, normal 0.75) high borehole inclination (60 ~ to 90 ~ neglecting effect bedding plane approach solving problem motion flyor is to solve following system equations inclination can give lower values fracture initiation pressure. However, for low borehole governing flow field detonation products behind flyor (Fig. I): inclination (0 ~ to 50 ~ neglecting effect bedding plane inclination will introduce only small error. 5.2 Shear failure in inclined boreholes ap +u_~_xp + au Shear failure produces an inward au movement au 1 plane failure; this usually happens when re is a drastic reduction internal borehole pressure such as unexpected loss drilling mud during production drawdown. One as is generally as concerned with shear failure mode since such failure mode can produce eir progressive collapse causing hole clean up problems or catastrophic collapse resulting in bridging sticking drill string potentially complete loss borehole. where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, Effect with degree trajectory R anisotropy reflected shock detonation wave D as a boundary trajectory Figure F 8 shows flyor as anor effect boundary. degree Both anisotropy are unknown; on collapse position pressure R as a function state para- meters borehole on inclination it are governed. A total by flow 5 curves field are I plotted central ranging rarefaction from wave n = behind 0.25 ( highly detonation anisotropic wave D by initial stage motion flyor also; position F state parameters products material) to n = (isotropic material), where n is degree anisotropy as defined in Eq. (39). From Fig. 8, it is observed that, in general, collapse pressure increases with increasing borehole inclination up to 50 degrees. For borehole inclination higher than 50 degrees, collapse

18 854 Dinesh Gupta Mushamff Zaman ~ 1.0.o 0.5 : :n =0.75] o o on =0.25 I 9... a Isotropic[ v.-'n = 0.50 I The one-dimensional problem n on =0.35] motion a rigid flying plate under explosive attack has an analytic solution only when polytropic index detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock 0 behavior reflection 0 shock in 20 explosive 40 products, 60 applying 80 small 90 parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying Boreholc inclination Angle fib/(*) plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities flying plate obtained ( 1 psi agree = very x well 10Spa, with 1 ft numerical = era) results by computers. Thus an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic Fig. 8 Effect degree anisotropy (Shear failure) index) for estimation velocity flying plate is established. pressure becomes fairly constant till it reaches horizontal plane. It can be seen that difference between isotropic anisotropic case 1. (n = Introduction 0.75) is not significant. From above analysis, one can conclude that material anisotropy has small effect on Explosive driven flying-plate technique ffmds its important use in study behavior materials borehole under collapse intense pressure impulsive at low loading, borehole shock inclinations. synsis Thus, diamonds, when degree explosive welding anisotropy is cladding low (n = metals. 1.00, The 0.75), method material estimation anisotropy flyor can velocity be neglected in way prediction raising it are questions collapse pressure common value interest. regardless borehole inclination Under Effect assumptions horizontal one-dimensional stress ratio plane detonation rigid flying plate, normal approach solving problem motion flyor is to solve following system equations The effect horizontal stress ratio (M) on stability borehole is shown in Fig. 9. It is governing flow field detonation products behind flyor (Fig. I): observed that for degree anisotropy (n = 0.75), difference between isotropic anisotropic solutions increases with increasing borehole inclination. There are certain critical borehole inclinations (35 ~ to 65 ~ ap for +u_~_xp low stress + au ratios (M = 0.75) where borehole fails regardless internal weubore pressurization. au au This 1 uncontrollable shear failure is caused by condition at borehole wall where stress state is such that failure criterion is always satisfied. In order to determine as change as in pressure values between degree borehole inclination, finer increments borehole inclinations ( in order 0.01 degrees ) should be used in computer program. This way, one can see exact value borehole inclination at which collapse pressure values are dropping to zero. where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, From with above trajectory analysis, R it can reflected be concluded shock that, detonation in general, wave D shear as a boundary failure in highly trajectory inclined boreholes F flyor (beyond as anor 40 o) boundary. is greatly Both influenced are unknown; by difference position in R horizontal state para- stress meters magnitudes. on it are Physically governed this by suggests flow field that I highly central inclined rarefaction boreholes wave behind are more detonation susceptible wave to D by initial stage motion flyor also; position F state parameters products collapse when value stress ratio M is expected to be low. This means that stability borehole is enhanced as ah,,x approaches ah The effect degree anisotropy on borehole shear failure is insignificant for lower borehole inclinations (0 ~ to 50 0).

19 Stability Boreholes in a Geologic Medium n =0.75 I.[ ~.0 ~0.5 m : :M = 0.90[ o o am = 0.75 The one-dimensional problem motion a rigid flying o plate om= under 1.00 explosive attack has an analytic solution only when polytropic index detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior reflection 0 shock in explosive products, applying small parameter purterbation method, an analytic, 0 first-order 20 approximate 40 solution 60 is obtained for 80 problem 90 flying plate driven by various high explosives with Borehole polytropic inclination indices fib/(~ or than but nearly equal to three. Final velocities flying plate obtained ( 1 psi = agree very x 103pa, well with 1 ft numerical = em) results by computers. Thus an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic Fig. 9 Effect horizontal stress ratio (Shear failure analysis) index) for estimation velocity flying plate is established Effect overburden-horizontal stress ratio In Fig. 10, collapse pressure 1. is Introduction plotted as a function borehole inclination for overburden-horizontal stress ratio V , It is observed that isotropic anisotropic Explosive solutions driven are flying-plate almost identical technique even ffmds at higher its important borehole use inclinations in study (beyond behavior 75 ~ For materials under intense impulsive loading, shock synsis diamonds, explosive welding higher stress ratio (V = 1.00), collapse pressure remains fairly constant at all borehole cladding metals. The method estimation flyor velocity way raising it are questions inclinations common interest. but for V = 0.90 V = 0.75, collapse pressure decreases with increasing borehole Under inclination. assumptions Specifically, one-dimensional for stress ratio plane V = detonation 0.75 decrease rigid in collapse flying plate, pressure is normal rar approach drastic beyond solving 55 ~ borehole problem inclination. motion flyor is to solve following system equations governing The above flow observation field detonation provides products some insight behind into flyor effect (Fig. I): overburden-horizontal stress ratios on shear failure pressure for a borehole. It can be concluded from above analysis that, in general, collapse pressure decreases with increasing borehole inclination decreasing ap +u_~_xp + au overburden-horizontal stress ratio. The effect anisotropy is found to be negligible in this case. Therefore, neglecting material anisotropy would not y introduce =0, significant error in prediction collapse pressure value. as as Effect in-situ stress orientation From Fig. 11, it is observed that, in general, for low stress inclination (/3~ = 0 ~ 30 o), collapse pressure increases up to 60 degree borehole inclination n starts decreasing for where higher p, borehole p, S, u are inclination. pressure, density, For stress specific inclination, entropy fl~ = particle 60 o, velocity pressure detonation decreases up products to 35 respectively, degree borehole with inclination trajectory R n it reflected increases shock for borehole detonation inclination wave D beyond as a boundary 35 degrees. For trajectory stress inclination, F flyor fls as = anor 90~, boundary. pressure keep Both on are decreasing unknown; with position increasing borehole R inclination. state parameters on it are governed by flow field I central rarefaction wave behind detonation wave It can be concluded from above analysis that, in general, collapse pressure increases D by initial stage motion flyor also; position F state parameters products for low stress inclination lower borehole inclination it decreases for higher stress inclination lower borehole inclination.

20 856 Dinesh Gupta Musharmf Zaman 1.5 v -vv = 0.75] o ov = 0.00[ = =V = 1.00~ n =0.75 "~' The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic solution only when polytropic index detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior reflection shock in explosive products, applying small parameter purterbation method, an analytic, first-order 20 approximate 40 solution 60 is obtained for 80 problem 90 flying Borehole inclination /~(~ plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities flying plate obtained ( 1 psi = agree very x 10~Pa,1 well with ft numerical = cm) results by computers. Thus an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic Fig. 10 Effect overburden-horizontal index) for estimation velocity flying plate is established. stress ratio (Shear failure) 1. Introduction 1.5 c ~ P, =30~ n =0,75 Explosive driven flying-plate =, ~,=90 technique ~ ffmds its important use in study behavior materials under intense impulsive v '~,8, loading, = 60 ~ shock synsis diamonds, explosive welding o o p,=o ~ cladding metals. The method estimation flyor velocity way raising it are questions common interest. Under 9 assumptions [ 1.o one-dimensional plane detonation rigid flying plate, normal \ approach solving problem motion flyor is to solve following system equations governing flow field detonation products behind flyor (Fig. I): -~ 0.5 ap +u_~_xp Borehole inclination/~b/(~ where p, p, S, u are pressure, density, specific entropy particle velocity detonation products ( 1 psi = x 103Pa, 1 ft = era) respectively, with trajectory R reflected shock detonation wave D as a boundary trajectory F flyor as Fig, anor 11 Effect boundary. in-situ Both are stress unknown; orientation position { Shear failure) R state parameters on it are governed by flow field I central rarefaction wave behind detonation wave D by Effect initial stage formation motion bedding flyor also; plane position orientation F state parameters products Figure 12 shows effect formation bedding plane inclination on collapse pressure as a function borehole inclination. au as as

21 Stability Boreholes in a Geologic Medium n =0.75 -L 1.0 Q Isotropic v ~ #,=60 ~ 0 0,8,. = 30 ~ 0 0:8,=~ Results indicate, in general, borehole 1. Introduction collapse pressure increases up to 50 degree borehole inclination it starts decreasing for borehole inclination beyond 50 degrees. It can be seen that Explosive driven flying-plate technique ffmds its important use in study behavior difference between isotropic anisotropic solutions are small it increases with increasing materials under intense impulsive loading, shock synsis diamonds, explosive welding cladding borehole inclination. metals. The method estimation flyor velocity way raising it are questions common It can interest. be concluded from above analysis, that for degree anisotropy ( n = 0.75) high Under bedding plane assumptions inclination, one-dimensional effect bedding plane detonation plane inclination rigid is flying not so plate, pronounced normal approach solving problem motion flyor is to solve following system equations neglecting it may introduce only small errors. governing flow field detonation products behind flyor (Fig. I): The results presented here show clearly effect different parameters on stability a borehole. Here range anisotropy used is between n = 0.25 to n = 1.00 which can be considered as representative actual ap +u_~_xp field conditions. + au Aadnoy's (1987 [~] ) results were based on values n = 0.71 n = au 0.63 which au cannot 1 be considered as representative many practical situations. Also, Aadnoy (1987 It] ) did not study effect anisotropic stress distribution around borehole which as have as been found to influence stability borehole significantly. The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic solution only when polytropic index detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior reflection 0 shock in 20 explosive 40 products, 60 applying 80 small 90 parameter purterbation method, an analytic, first-order approximate solution is obtained for problem flying Borehole inclination /3b/(~ plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities flying plate obtained 1 psi = agree very x 103Pa,1 well with ft numerical = cm) results by computers. Thus an analytic formula with two parameters Fig. 12 Effect high explosive formation (i.e. bedding detonation plane velocity polytropic index) for estimation velocity flying plate is established. inclination ( Shear failure} 6 Analysis Plastic Zone Around a Borehole where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, Elasticity with may be trajectory a good R idealization reflected for shock a ~eologic detonation medium wave for D certain as a boundary cases involving trajectory relatively F small flyor deformations, as anor however, boundary. it Both is not are an unknown; effective idealization position in many R or state instances. parameters Idealization it are rocks governed as an by elasto-plastic flow field materialis I central probably rarefaction a more wave realistic behind approach. detonation In wave case D by initial stage motion flyor also; position F state parameters products borehole stability problems, it is appropriate to consider a plastic zone surrounded by elastic region. In order to underst stability a borehole in an elasto-plastic formation, it is necessary to study plastic behavior material possible stress distributions caused

22 858 Dinesh Gupta Musharraf Zaman by plastic flow. The analytical approach employed here is similar to that Infante Chenevert (1989 [~s] ), however, it was extended to include anisotropy material. The ories elasticity plasticity are used to calculate distribution stresses around a borehole. The formation was assumed to be elastic-perfectly plastic without any strain hardening stening behavior taken into account. The mamatical formulation for both elastic zone plastic zone is developed in this study. 6.1 Model description For convenience in mamatical description problem, a vertical cylindrical hole through a horizontal layer a transversely isotropic geologic material is assumed. Figure 13 shows an infinitely long thick walled cylinder ge- Ro ologic The material one-dimensional with inner radius, problem R/e, motion outer ra- a rigid flying plate under explosive attack has an dius, analytic Ro, located solution in a only hydrostatic when stress polytropic field. Since index detonation products equals to three. In general, formation a numerical material considered analysis is here required. exhibits In elasticthis paper, however, by utilizing "weak" shock behavior reflection shock in explosive products, applying small parameter purperfectly plastic behavior, a boundary between terbation method, an analytic, first-order approximate solution is obtained for problem flying plate two modes driven by behavior various high exists explosives when with stress polytropic field indices or than but nearly equal to three. ~ -Plastic Final around velocities a borehole satisfies flying plate octahedral obtained agree shear very stress well with numerical results by computers. undary Thus an ory analytic (I.rffante formula Chenevert, with two parameters 19891ts]). Let high Rp be explosive (i.e. detonation velocity polytropic index) radius for estimation boundary as velocity shown in Fig. flying 13. plate Com- is established. pressive stresses are considered positive. The formation is assumed to be transversely isotropic, 1. homoge- Introduction Fig. 13 Geometry posed' problem neous continuous. Explosive driven flying-plate technique ffmds its important use in study behavior 6.2 Basic equations materials under intense impulsive loading, shock synsis diamonds, explosive welding cladding To calculate metals. The state method stress estimation around a bore- flyor velocity way raising it are questions hole, common following interest. relationships must be satisfied (Hsiao, 1988 C71 ): Under 1) Equations assumptions equilibrium; one-dimensional plane detonation rigid flying plate, normal approach 2) Elastic solving stress-strain problem relations; motion flyor is to solve following system equations governing 3) Strain-displacement flow field detonation relations products ; behind flyor (Fig. I): 4) Plastic flow rule; 5) Octahedral shear stress yield function; ap +u_~_xp + au 6) Boundary conditions. The octahedral shear stress yield function is given y as =0, follows: F = 4(O'r -- 0"0) 2 as + (0-r as -- o-z) 2 + (0"0 -- o-z) r. = 0, where ro being octahedral shear stress, a r, a0 az are principal stresses in radial, tanp =p(p, s), gential vertical direction, respectively. where The p, p, elastic S, u are stress pressure, solution density, can be specific written entropy as follows: particle velocity detonation products respectively, with trajectory R reflected shock detonation wave D as a boundary trajectory F flyor as anor boundary. 0-r "" 0-f- Both (0-f- are unknown; Pw)(-~) 2, position R state (43a) parameters on it are governed by flow field I central rarefaction wave behind detonation wave D by initial stage motion flyor also; position F state parameters products (42) ve. a, = 2af--~-, (43c)

23 Stability Boreholes in a Geologic Medium 859 where a r, ao az are principal stresses in radial, tangential vertical direction, respec- tively, af is hydrostatic stress around borehole. E is Young' s modulus in horizon- tal plane, E z is Young' s modulus in vertical direction, v is Poisson' s ratio in horizontal plane. The above relation gives elastic state stress after drilling borehole. The elastic stress solution given above is valid only until yield condition given by Eq. (42) i~ satisfied. In particular, plastic yielding is first initiated at borehole wall, i. e. at r = Rw. Inserting r = Rw in elastic stress solution, one obtains following expressions: O'r : Pw, (44a) a0 = 2af- Pw, (44b) cr~ = 2o'f vez E " (44c) The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic solution only when polytropic index detonation products equals to three. In Combining Eqs. (44a), (44b) (44c) with Eq. (42), one can predict borehole presgeneral, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior sure that initiates reflection plastic flow shock by in solving explosive for only products, one unknown, applying Pw combined small parameter equation. purterbation Once method, plastic an flow analytic, initiates, first-order it grows approximate with increasing solution borehole is obtained pressure. for The problem rock formation flying plate around driven a borehole by various is n high separated explosives into with a plastic polytropic zone indices ( Rw < or r < Rp) than but an nearly elastic equal zone to ( three. Rp < Final r < velocities Ro) as shown flying in Fig. plate 13. obtained The state agree very stress well in with plastic numerical zone results must satisfy by computers. conditions Thus an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic equilibrium, plastic flow rule octahedral shear stress yield function. The stress soindex) for estimation velocity flying plate is established. lution for plastic zone in elasto-plastic formation is given as follows: 1. Introduction err = Pw + 3to In, (45a) Explosive driven flying-plate technique ffmds its important use in study behavior materials under intense impulsive loading, shock synsis diamonds, explosive welding cladding metals. The method go = P,,+ estimation 3ro~/(2-~)[ln(~ flyor velocity )- 1], way raising it are questions (45b) common interest. Under assumptions one-dimensional plane detonation rigid flying plate, normal approach solving problem az - E motion + flyor is to solve following - ' system equations governing where at, ao flow field az are detonation principal products stresses behind plastic flyor zone (Fig. around I): borehole. K' is a constant given by ap +u_~_xp + au The plastic region usually does not extend to infinity, because beyond a certain distance from borehole, at r = Rp, stresses as are as such magnitude that elastic behavior is present. The final expressions for elastic stresses in elasto-plastic formation can be written as follows (Gupta, 1994 [191) : where p, p, S, u are O'r pressure, = 0"f--( density, R-~-P-r )2{or specific f - p,,+ entropy }ro~/(2@)[in particle velocity ( RR--~)] }, detonation products (47a) respectively, with trajectory R reflected shock detonation wave D as a boundary trajectory F flyor as anor boundary. Both are unknown; position R state parameters on it are governed ao = ~f+ by (R---~r)2Iar- flow field I P~+3ro~/(~-TK,)[ln(~-~)] central rarefaction wave behind 1, detonation (47b) wave D by initial stage motion flyor also; position F state parameters products ve~ a~ = 2af E. (47c)

24 860 Dinesh Gupta Musharraf Zaman 6.3 Extent plastic zone The expression for radius plastic zone, Rp, can be obtained by combining Eqs. (45a), (45b), (47a) (47b) inserting r = R v, as follows: 3 1 Rp = Rwexp{[-fro~2~ + a t- Pw]/3ro~}. (48) In above analysis, a mamatical formulation is derived to calculate stress distribu- tion around a borehole in an elastic medium, state stress at which plastic flow is initiat- ed, stress distribution in plastic zone radius plastic zone. 6.4 Failure criterion for plastic zone In formations exhibiting elastic-perfectly plastic behavior, no strain hardening or sten- ing, failure criterion can be expressed in terms growth plastic zone in forma- The one-dimensional problem motion a rigid flying plate under explosive attack has tion (Hsiao, ]). Increasing ratio plastic zone tends to reduce elastic region, an analytic solution only when polytropic index detonation products equals to three. In general, to reduce a numerical borehole analysis stability. is required. If In plastic this paper, zone however, propagates by throughout utilizing "weak" whole shock formabehavior tion, very large reflection deformations shock are in expected explosive to occur products, formation applying will collapse. small parameter A nondimen- purterbation sional plastic method, zone an ratio analytic, (borehole first-order stability approximate ratio) can solution n be determined is obtained for to indicate problem conditions flying plate driven formation by various failure. high This explosives quantity with (borehole polytropic stability indices ratio) or is defined than but as nearly ratio equal to three. plas- Final velocities flying plate obtained agree very well with numerical results by computers. Thus tic zone radius to outer radius model. an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic index) 6.5 for Discussion estimation results velocity parametric flying plate study is established. for plastic zone The mamatical formulation developed in preceding section is used to evaluate stress distribution around a borehole drilled in a transversely isotropic medium displaying plastic 1. Introduction behavior. In order to investigate wellbore stability problem in an elasto-plastic formation, influence Explosive different driven parameters flying-plate that technique may affect ffmds its stress important distribution use in around study a borehole behavior sta- materials bility under borehole intense is examined. impulsive loading, shock synsis diamonds, explosive welding cladding In its metals. present The form, method formulation estimation can be flyor used velocity only to model a way vertical raising borehole it are situated questions in common interest. a hydrostatic stress field. Parameters such as type material anisotropy (isotropic transverse- Under assumptions one-dimensional plane detonation rigid flying plate, normal approach ly isotropic), solving degree problem anisotropy, motion in-situ flyor stress is values to solve ir following effects system on wellbore equations stability governing stress distribution flow field are detonation investigated products based on behind proposed flyor formulation. (Fig. I): A computer program is develope~ to numerically, evaluate, distribution stresses around a borehole, to predict borehole pressure required to initiate plastic flow to predict critical pressure at which ap +u_~_xp + au formation around borehole would fail. The program f'u'st calculates au elastic au stresses. 1 Once stress sate satisfies octahedral shear stress yield function, plastic flow is initiated. As borehole pressure is increased, plastic zone propagates When borehole as pressure as reaches critical valur that causes plastic zone to cover entire formation, formation is said to have failed. The effect following parameters are investigated: where 1) p, Degree p, S, u are Anisotropy; pressure, density, specific entropy particle velocity detonation products respectively, 2) In-Situ with Stress trajectory Value. R reflected shock detonation wave D as a boundary trajectory Effect F flyor degree as anor boundary. anisotropy Both are unknown; position R state parameters In on this it are analysis, governed isotropic by material flow field I transversely central rarefaction isotropic wave materials behind are considered. detonation wave The D by initial stage motion flyor also; position F state parameters products degree anisotropy, n, is defined by Eq. (39). Figures 14, 15, 16 show stress distribution in an elasto-plastic formation for iso tropic case for degrees anisotropy n = n = 0.6 0, respectively. The borehole pressure at which plastic flow is initiated,

25 Stability Boreholes in a Geologic Medium x 104 lsotropic... Vertic'alstresses... Tangential stresses Radial stresses //./...%... 9 _... l.s lo' x 104 //"~''~'~"... ' j,,~"... / /./ [ //./ The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic solution only when polytropic index detonation products equals to three. In general, a numerical analysis 0 is required. 2.5 In this 5.0 paper, however, 7.5 by utilizing I0.0 "weak" shock behavior reflection shock in explosive products, applying small parameter purterbation method, an analytic, first-order approximate Nondimensional solution radius ( r/rw is obtained ) for problem flying plate driven by various high explosives (n with = 1.00) polytropic (1 psi indices = or x 103Pa) than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus Fig. 14 Stress distribution in dasto-plastic an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic index) for estimation velocity flying formation plate is established. around a borehole Introduction I In i J... Vertical stresses 9 "'" I... ~ Tangential stresses Explosive driven 1.5x flying-plate 104 :/./-" """ technique... ffmds its I important... Radial use stresses in study behavior materials under intense impulsive loading, shock synsis diamonds, explosive welding e~ cladding metals. The method estimation flyor velocity way raising it are questions common interest. Under :3 e~ assumptions! one-dimensional plane detonation rigid flying plate, normal /,/ approach solving problem motion flyor is to solve following system equations ': /11 "'~"~. governing flow field detonation products behind flyor (Fig. I): I.I J"...! // 0.5x104 } t" ap +u_~_xp + au as as , Nondimensional radius (r/r,,) (n = 0.75) (1 psi = x 103Pa) where p, p, S, u are pressure, density, Fig. 15 specific Stress entropy distribution particle in elasto-plastic velocity detonation products respectively, with trajectory R reflected shock detonation wave D as a boundary trajectory F flyor as anor boundary. formation Both are around unknown; a borehole position R state parameters on it are governed by flow field I central rarefaction wave behind detonation wave D extent by initial plastic stage zone motion stress flyor distributions also; in position elastic F plastic state zone parameters are shown products in se figrues. It is observed that relative positions curves change with degree anisotropy. The isotropic anisotropie solutions are compared percentage difference in radius plas-

26 862 Dinesh Gupta MusharrafZaman 2.0 x 104 b n = Vertical stresses l... Tangential stresses /^\., Radial stresses // \'~... /.g r l.ox I04 0.5x 104,/ /" /./ / / i /s ~... t / The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic solution only when polytropic index detonation products equals to three. In 0 general, a numerical analysis 1.0 is required. 2.5 In this 5.0 paper, however, 7.5 by utilizing 10.0 "weak" shock behavior reflection shock in explosive products, applying small parameter pur- Nondimensional radius ( r/r~ ) terbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven by various high explosives (n with = polytropic 0.60)(1 psi indices = or x 103Pa than ) but nearly equal to three. Final velocities flying plate Fig. obtained 16 agree Stress very distribution well with in numerical elasto-plastie results by computers. Thus an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic formation around a borehole index) for estimation velocity flying plate is established n = 0.60] 1. Introduction... n = 0.75 I Explosive driven flying-plate... Isotropic technique ] ffmds its important use in study behavior 1.00 j.:s: materials under intense impulsive loading, shock synsis diamonds, S~. r explosive welding cladding metals. The method estimation flyor velocity way raising it are questions common interest. o N 0.75 o Under assumptions one-dimensional plane detonation rigid flying plate, normal approach solving problem motion flyor is to solve 4g.'" following system equations ~.~" governing flow field detonation products behind flyor -g 0.50 ~,. (Fig. I): o z x 104 ap +u_~_xp +./ 1.0x l04 as 1.5x as I04 2.0x x x 104 Borehole pressure (P,,/psi) (1 psi = x 103Pa ) where p, p, S, u are pressure, density, Fig. 17 specific Effect entropy degree particle anisollopy velocity detonation products respectively, with trajectory R reflected shock detonation wave D as a boundary on stability ratio trajectory F flyor as anor boundary. Both are unknown; position R state parameters on it are governed by flow field I central rarefaction wave behind detonation wave D tic zone by between initial stage isotropic motion case flyor n also; = 0.60 position case, is F found to be state parameters % which products is quite large. At boundary between plastic zone elastic zone, tangential stresses must be con- tinuous. From expressions for radial stressses in elastic zone plastic zone, it is clear that au

27 Stability Boreholes in a Geologic Medium 863 re will be a reduction in radial stress values after elasto-plastic boundary whereas expres- sions for tangential stress indicates that stress values will increase with increase in radial dis- tance in both zones. This explains continuous nature curve at elasto-plastic boundary for tangential stresses. The borehole pressure required to initiate plastic flow is higher for a high degree anisotropic rock as compared to isotropic rocks. The percentage difference in borehole pressure required to initiate plastic flow, between isotropic anisotropic ( n = 0.60) cases, is found to be 23.80%. Figure 17 shows growth plastic zone with increasing borehole pressure for degrees anisotropy n = 0.75, 0.60 results are compared with isotropic solu- tion. Results indicate that critical borehole pressure required to reach a stability ratio one is not significantly influenced by degree anisotropy. The percentage difference in critical borehole pressure The for one-dimensional a high degree problem anisotropy, motion n = 0.60, a rigid flying isotropic plate solution under explosive is 2.12% attack which has is an very analytic small. solution only when polytropic index detonation products equals to three. In general, The a numerical above observations analysis is give required. some insight In this into paper, effect however, degree by utilizing anisotropy "weak" on shock stress behavior reflection shock in explosive products, applying small parameter purdistribution in elastic plastic zones, borehole pressure yielding occurs stability terbation method, an analytic, first-order approximate solution is obtained for problem flying plate borehole. driven by It various can be high concluded explosives that with degree polytropic anisotropy indices or is an than important but nearly parameter equal to three. ne- Final glecting velocities would give flying erroneous plate obtained prediction agree very borehole well with collapse numerical pressure. results by computers. Thus an analytic Effect formula with overburden two parameters in-situ stress high explosive value (i.e. detonation velocity polytropic index) The for estimation effect overburden velocity in-situ flying stress plate value is established. on stability ratio as a function borehole pressure is shown in Fig. 18. Results are plotted for overburden stresses psi ( 1 psi = x 103 Pa) psi isotropic 1. Introduction anisotropic (n = 0.75) solutions are compared. The formation is assumed to have a overburden stress gradient 1.0 psi/ft depth Explosive driven flying-plate technique ffmds its important use in study behavior borehole is assumed to be ft (1 ft = cm) ft. From Fig. 18, it is materials under intense impulsive loading, shock synsis diamonds, explosive welding cladding observed that metals. borehole The method pressure estimation required to flyor initiate velocity plastic flow way increases raising with it are an questions increasing common interest Under assumptions... one-dimensional n =0.75, Hydro. stress= plane 12 detonation 500 psi rigid flying plate, normal approach solving problem... n = 0.75, motion Hydro. stress flyor = 10 is 000 to solve psi following system equations ~=.1.00, Hydro. stress = psi t: /7 governing flow field detonation products behind flyor (Fig. I):... n = 1.00, Hydro. stress= psi z// - // ff //,'," /,// /, 0.75 ap +u_~_xp + au Z//r,~, O ".~ o~ 0.50._ 0.25 where p, p, S, u are pressure, 0 density, specific entropy particle velocity detonation products respectively, with trajectory 0 R reflected I I0 ~ shock detonation wave D as 3 a boundary I0 4 trajectory F flyor as anor boundary. Both Borehole are pressure unknown; (P./psi) position R state parameters on it are governed by flow field I central rarefaction wave behind detonation wave D by initial stage motion flyor (1 also; psi = position 103Pa) F state parameters products Fig. 18 as as Effect in-situ overburden stress on stability ratio

28 864 Dinesh Gupta Musharraf Zaman value hydrostatic stress with higher degrees anisotropy. In general, critical boreholepressure required for failure whole formation is higher for higher in-situ stress value. For example, in case an isotropic geologic medium, critical pressure required for overburden stress value equal to psi is about 9% higher than critical pressure required for overburden stress value equal to psi. Also, it is observed that percentage difference between values critical borehole pressure, for isotropic case anisotropic case ( n = 0.75), is about 1.20% which is very small. It can thus be concluded from above observations that degree anisotropy has a very small effect on critical borehole pressure required to cause failure formation. Overburden in-situ stress value has a pronounced effect on critical borehole pressure neglecting it can introduce errors in prediction critical pressure value. The one-dimensional problem motion a rigid flying plate under explosive attack has an 7 analytic Summary solution only Conclusions when polytropic index detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock 1. An approximate, three-dimensional anisotropic elastic model for evaluating stress behavior reflection shock in explosive products, applying small parameter purterbation distribution method, around an a analytic, borehole first-order drilled in approximate an anisotropic solution geologic is obtained medium for has been problem developed. flying The plate orientation driven by various borehole, high explosives in-situ with stresses polytropic indices bedding or than plane but inclination nearly equal can to three. all be Final arbitratily velocities related flying to each plate or obtained to model agree actual very field well conditions. with numerical results by computers. Thus an analytic 2. The formula model with utilizes two minimum parameters normal high stress explosive ory (i.e. for tensile detonation failure velocity analyses polytropic Druckerindex) for estimation velocity flying plate is established. Prager failure criterion is used for shear failure analyses. 3. Based on mamatical formulation, two computer programs were developed, one for 1. Introduction fracture initiation pressure or one for collapse pressure. These programs provide an effective Explosive means driven for evaluating flying-plate technique sensitivity ffmds various its important parameters use that in affect study stability behavior a materials borehole. under intense impulsive loading, shock synsis diamonds, explosive welding cladding 4. The metals. effect The anisotropy method on estimation fracture initiation flyor velocity pressure becomes way significant raising it only are in questions case common highly inclined interest. boreholes. The difference between isotropic anisotropic cases is small for Under assumptions one-dimensional plane detonation rigid flying plate, normal low borehole inclination (between 0 35 degrees) adn isotropic solutions can be used for approach solving problem motion flyor is to solve following system equations governing analysis. flow field detonation products behind flyor (Fig. I): 5. The effect in-situ stress conditions becomes significant in cases highly angled wells (above 40 degree inclination) a high degree anisotropy (n = 0.75). Low ratios horizontal stresses overburden-horizontal ap +u_~_xp stresses + au can affect borehole stabiliy significantly. 6. For low borehole inclination au (up au to 401 degrees ), effect hte bedding plane inclination can be neglected, but for a higher borehole inclinations a higher degree anisotropy, neglecting effect as beeding as inclination can result in less conservative values fracture initiation pressure. 7. It was found that collapse pressure increases with increasing borehole inclination. The where material p, p, anisotropy S, u are pressure, can be neglected density, specific in case entropy collapse particle pressure velocity analysis detonation use products respectively, isotropic solution with may trajectory be justified. R reflected shock detonation wave D as a boundary trajectory 8. The F collapse flyor as pressure anor boundary. is greatly Both influenced are unknown; by difference position in R horizontal state para- stress meters on it are governed by flow field I central rarefaction wave behind detonation wave values. The difference between isotropic anisotropic solutions becomes significant only for a D by initial stage motion flyor also; position F state parameters products high degrees anisotropy high borehole inclinations (above 50 degrees). 9. For high degrees anisotropy high bedding inclinations, neglecting effect bedding plane inclination can cause errors in predicting collapse pressure values.

29 Stability Boreholes in a Geologic Medium The stability a borehole in an elasto-plastic medium is also investigated. The mamatical formulation developed by Infante Chenevert (1989 [18] ) was modified to include anisotropy material. The formation was assumed to be elastic-perfectly plastic case a vertical borehole was analyzed. The formulation was developed using ories elasticity plasticity. A computer program was developed to evaluate sensitivity different parameters on development plastic zone around a borehole in an anisotropic geologic medium. 11. The effect material anisotropy overburden in-situ stress values was investigated on plastic zone. It was found that material anisotropy overburden in-situ stress values influence critical collapse pressure values for a borehole. In general, it can be concluded that fracture initiation pressure collapse pressure in an inclined The boreholes one-dimensional are significantly problem influenced motion by a a hogh rigid flying degrees plate under borehole explosive inclination, attack high has an degree analytic material solution anisotropy, only when high polytropic formation index bedding plane detonation inclinations products equals in-situ to three. stress In general, conditions. a numerical The significant analysis finding is required. this In study this paper, is that however, borehole by stability utilizing is dependent "weak" on shock behavior anisotropic properties reflection shock geologic in medium explosive products, in-situ stress applying conditions around small parameter borehole. purterbation method, an analytic, first-order approximate solution is obtained for problem flying plate Acknowledgements: driven by various high Authors explosives would with like polytropic to thank indices Dr. Ong or S. than H. for but his nearly useful equal suggestions to three. Final comments. velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic index) References: for estimation velocity flying plate is established. [ 1 ] Aadnoy B S. Continuum mechanics analysis stability boreholes in anisotropic rock 1. Introduction formations[d]. Ph D Thesis. Norway: Norwegian Institute Technology, University Explosive Trondheim, driven flying-plate technique ffmds its important use in study behavior materials E21 Ong under S intense H. Borehole impulsive stability loading, [ D shock ]. Ph synsis D Dissertation. diamonds, Norman explosive OK: University welding cladding Oklahoma, metals The method estimation flyor velocity way raising it are questions [3] common Lekhnitskii interest. S G. Theory Elasticity an Anisotropic Body [ M I. Moscow: Mir Under assumptions one-dimensional plane detonation rigid flying plate, normal Publishers, approach solving problem motion flyor is to solve following system equations governing [4] Amader flow B. field Rock anisotropy detonation products ory behind stress flyor measurements (Fig. I): [ A]. In: Brebbia, Orszag Eds. Lecture Notes in Engineering[C]. Springer Verlag, [5] Aadnoy B S. A complete elastic model for fluid-induced in-situ generated stresses with presence a borehole[ J]. ap Energy +u_~_xp Sources, + au 1987,9 : 239 ~ 259. E6] Bradley W B. Failure inclined au boreholes[j]. au 1 y Journal =0, Energy Resources Technology Trans, ASME, 1979,101:232 ~ 239. [7] Hsiao C. Growth plastic as zone in as porous medium around a weubore [ J ]. Offshore Technology Conference, 1988,26: [81 p Westergaard H M. Plastic state =p(p, stresses s), around a deep well[jl. Journal Boston where Society p, p, S, u are Civil pressure, Engineers, density, 1940,27( specific 1 ) : entropy 1 ~ 5. particle velocity detonation products respectively, [9] Gnirk with P F. The trajectory mechanical R behavior reflected shock uncased wellbores detonation situated wave D in as elastic/plastic a boundary media trajectory under F hydrostatic flyor as anor stress[j]. boundary. Society Both Petroleum are unknown; Engineers position Journal, 1972,12:49 R state parameters [iol on Risnes it are R, governed Bratli by R K, flow Horsud field P. I S central stresses rarefaction around wave a behind wellbore [ detonation J ]. Society wave D by initial stage motion flyor also; position F state parameters products Petroleum Engineers Journal, 1982,22:883 ~ 898. [III Biot M A. Theory elasticity consolidation for a porous anisotropic solid [ J ]. Journal Applied Physics, 1941,26(2) : 182 ~ 185.

30 [12] [13] [14] [15] 866 Dinesh Gupta Musharraf Zaman materials[ J]. Journal Engineer Mechanics Division, ASCE, 1986,112: 1041 ~ [16] Najjar Y M. Constitutive modelling finite element analysis ground subsidence due to mining[d]. Ph D Thesis. Norman OK: University Oklahoma, [17] The Lama one-dimensional R D, Vutukuri problem V S. Hbook motion on Mechanical a rigid flying Properties plate under Rocks[M]. explosive attack Series has on an analytic Rock solution Soil only Mechanics when Vol. polytropic 1/, Ohio: index Trans Tech detonation Publications, products equals to three. In general, [18] Infante a numerical E F, analysis Chenevert is required. M E. In Stability this paper, boreholes however, drilled by utilizing through salt "weak" formations shock behavior reflection shock in explosive products, applying small parameter purdisplaying plastic behavior [ J ]. Society Petroleum Engineers Drilling Engineering terbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven ( SPEDE), by various 1989,4: high explosives 57 ~ 65. with polytropic indices or than but nearly equal to three. Final [19] velocities Gupta D J. flying Stability plate Of obtained boreholes agree in very a transversely well with isotropic numerical geologic results by medium[d]. computers. Thus M S an analytic Thesis. formula Norman with two OK: parameters University high Oklahoma, explosive (i.e. detonation velocity polytropic index) [20] for Aadnoy estimation B S. Stability velocity highly flying inclined plate is boreholes[j]. established. SPEDE, 1987,2:364 ~ 374. [21] Aadnoy B S. Modeling stability highly inclined boreholes in anisotropic rock formations[ J]. SPEDE, 1988,2: Introduction ~ 267. [22] Yew C H, Li Y. Fracture a deviated well[j]. SPE Production Engineering, 1988,3: 429 ~ 437. Chen Dar-Hao. Three dimensional testing constitutive modeling coal for analysis ground subsidence due to mining [ D ]. M S Thesis. Norman OK: University Oklahoma, Drucker D C, Prager W. Soil mechanics plastic analysis or limit design [ J]. Quart Appl Math, 1952,10:157 ~ 165. Faruque M O, Chang C. New cap model for failure yielding pressure sensitive Aadnoy B S. Method for fracture-gradient prediction for vertical inclined boreholes Explosive driven flying-plate technique ffmds its important use in study behavior [J]. SPEDE, 1989,4:99 ~ 103. materials under intense impulsive loading, shock synsis diamonds, explosive welding cladding [23] Aadnoy metals. B S. The Stresses method around estimation horizontal flyor boreholes velocity drilled in sedimentary way raising rocks[j]. it are questions Journal common Petroleum interest. Science Engineering, 1989,2: 349 ~ 360. [24] Under Aadnoy assumptions B S. Effects one-dimensional reservoir depletion plane on detonation borehole stability[j]. rigid flying Journal plate, Petroleum normal approach Science solving Engineering, problem 1991,5 motion : 57 ~ flyor 61. is to solve following system equations governing [25] flow field detonation products behind flyor (Fig. I): Bratli R K, Risnes R. Stability failure s arches [ J ]. Society Petroleum Engineers Journal, 1981,21:236 ~ 248. [26] Tsai S W, Wu E M. A general ap +u_~_xp ory + au strength for anisotripic materials [ J ]. J Composite Material, 1971,5 : au 58 ~ 80. au 1 as as where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, with trajectory R reflected shock detonation wave D as a boundary trajectory F flyor as anor boundary. Both are unknown; position R state parameters on it are governed by flow field I central rarefaction wave behind detonation wave D by initial stage motion flyor also; position F state parameters products