Borehole breakdown pressure with drilling fluids—II. Semi-analytical solution to predict borehole breakdown pressure

Int. J. RockMech. Min. Sci. & Geomech.Abstr.Vol. 33, No. I, pp. 53-69, 1996 Pergamon

Copyright © 1996 Elsevier ScienceLtd Printed in Great Britain. All fights reserved 0148-9062/96 $15.00 + 0.00

0148-9062(95)00029-1

Borehole Breakdown Pressure with Drilling Fluids II. Semi-analytical Solution to Predict Borehole Breakdown Pressure N. MORITAf G.-F. FUH~: A. D. B L A C K §

The fracture initiation and propagation experiments discussed in Part I revealed that three distinct zones exist around a fracture tip. They are the fracture process zone, the non-invaded and the dehydrated gel zone. These three zones significantly increase wellbore breakdown pressure and fracture propagation resistance under a relatively large in s i t u stress. With the assumption that wellbore pressure is sealed at the inlet o f the dehydrated mud zone, a semi-analytical model o f a fractured, inclined borehole is developed f o r a general plane strain condition. The model predicts that when drillingfluid is used as an injection fluid, borehole breakdown pressure is highly dependent on the formation Young's modulus, the in situ stress, the wellbore size, and type o f the drilling fluids, although conventional theories have not included all these.

NOMENCLATURE

time dimensionless time defined in Appendix T,T~= temperature, temperature at wellbore W= fracture width WI, W2, W3, W4, Wp = fracture widths corresponding to the hydrostatic boundary stress, directional stress, borehole pressure, fracture surface pressure and pore pressure, respectively. Wm= fracture width at the sealing point with mud cake Wt = fracture width at the point which does not allow the mud invasion W0 = fracture width of a pre-existing fracture ~o = defined in Table 3 ct = thermal expansion coefficient fl, 7 = r,/A, well angle a = radial stress ao= tangential stress a z = axial stress tYHi , Gh2 = in situ stresses (negative) tru, Ov= horizontal and vertical in situ stress Id =

A = well radius plus fracture length B = well radius plus the fracture interval invaded with drilling fluid c = fluid compressibility C = specific heat q, Cb= rock matrix and bulk compressibilities E, v = elastic moduli F I, F2, F3 = constants given in Table 1 G 1, G2, G3 = constants given in Table 1 H I, H2, H3, H4 = constants given in Table 2 K = thermal diffusion coefficient Kc = fracture toughness Kf, KT = permeability, thermal diffusivity P, P . = pressure, well pressure pressure within a fracture ef pore pressure at the borehole surface eow q = pore pressure gradient around a borehole Qr = temperature gradient around a borehole r = radial coordinate r=

r/rw

GI ~

well radius R r = fracture radius up to the pressure sealing point R m : the length of dehydrated mud around the fracture tip R t = length of the fracture tip without mud invasion S, S1 = defined in Table 2

INTRODUCTION A standard borehole breakdown analysis uses the K i r s h ' s solution which provides the linear elasticity equation for a general plane strain condition. Borehole b r e a k d o w n p r e s s u r e is a s s u m e d t o o c c u r w h e n t h e

fDepartment of Mineral Resources Engineering, Waseda University, Ohkubo 3-4-1, Shinjuku, Tokyo, Japan. ~Conoco Inc., 1000 South Pine Street, Ponca City, OK 174602, U.S.A. §University Research Park, 400 Wakara Way, Salt Lake City, UT 84108, U.S.A. RUMS~3/,--E

GHI -- GI~

(, r/= 2~/A/n, B/A p, # = fluid density, viscosity ~b= porosity

rw

m a x i m u m tangential stress reaches the tensile rock strength. H o w e v e r , l a b o r a t o r y o b s e r v a t i o n s s h o w t h a t if 53

54

MORITA et al.: BOREHOLEBREAKDOWN--SEMIANALYTICALSOLUTION

drilling fluid is used as a fracturing fluid, the borehole breakdown does not occur even if an initiated fracture propagates as much as 0.76-7.62 cm (0.3-3 in.) [up to 10 cm (4 in.) at rock boundary] for laboratory conditions and depending on the confining pressure, the solid content of drilling fluid and formation Young's modulus. The fracture extension pressure initially increases with fracture length but, when the drilling fluid starts penetrating into the fracture due to the enlarged fracture aperture, the fracture extension pressure reduces, resulting in borehole breakdown. Since borehole breakdown pressure depends on the fracture width around the fracture tip, borehole breakdown pressure becomes dependent on the formatin Young's modulus, borehole size, confining pressure and the plugging capability of drilling fluid. The Kirsh 2-D plane strain solution considers an inclined borehole problem by transforming the in situ stress into components with the z-axis coinciding with the borehole axis. The in-plane in situ stress components are axx, ayy, trxy and a=. The out-of-plane in situ stress components azx and azy affect the deformation in the z direction but they do not affect the deformation in the x y plane. In addition, the outof-plane in situ stress components do not affect the tangential stress around a well. This fact significantly simplifies the general plane strain problem since if the two horizontal in situ stresses are equal, a mode I fracture is induced and propagates in the well azimuth direction and the width and propagation criterion are not affected by the out-of-plane in situ stresses. Hence, the stress intensity factor and width of the fracture induced along the well azimuth direction can be calculated by a 2-D fracture model instead of a 3-D fracture model, although the problem itself is a general plane strain problem with a 3-D deformation. To predict the borehole breakdown pressure observed in the laboratory scale and field scale wells, an analytical model to simulate a fracture extension around a borehole is needed. Since no analytical solution has been found for the stress intensity factor and fracture width for the crack from a borehole, an approximate solution is found by numerical experiments. First, a closed form solution is found for the fracture width and stress intensity factor for a planar crack without a borehole. Then, correction factors are added for the borehole effect such that the analytical equation asymptotically approaches the analytical solution without the borehole if the fracture becomes significantly large. The correction factors are determined by fitting to the results of the numerical experiments generated by a finite element model. Parameter analysis for borehole breakdown pressure was conducted using the semi-analytical solution. The semi-analytical solution predicts the effects of Young's modulus, confining stress and plugging of drilling fluids on borehole breakdown. The results match with the laboratory results reported in Part I. The semi-analytical solution is used to study the field observations of borehole breakdown pressures under various conditions.

SEMI-ANALYTICAL SOLUTION OF BOREHOLE BREAKDOWN PRESSURE

Figure 1 shows the boundary conditions to be solved. The well and part of the fracture (up to the sealing point of dehydrated drilling fluid) have pressure Pw and the pressure between the sealing point and the fracture front is set to equal to the pore pressure. Inclination of borehole, temperature and pore pressure effects are also taken into account. Among the loads (three in situ stresses, pore pressure, temperature, borehole pressure, pressure in cracks), only the in situ stress components include the out-of-plane stress components (out-of-plane in situ stress: axe, ayz in-plane in situ stress: axy, trxy, ayy, tr~z) which may affect the stress intensity factor and the fracture width. Two- and three-dimensional finite element meshes were constructed to study the out-of-plane stress effect as shown in Fig. 2. It was found that if the two horizontal in situ stresses are assumed to be equal or trm = trn2, the out-of-plane in situ stress does not affect the fracture width and the stress intensity factor for the following reasons: (A) A fracture initiated at the minimum tangential stress direction propagates as an in-plane fracture if O'HI a m . The fracture propagates as a mode I fracture propagation in the well azimuth direction. (B) Numerical experiments showed that out-of-plane stresses did not affect the stress intensity factor and fracture width if the fracture propagates as mode I propagation. Only the stress normal to the fracture plane and stress perpendicular to the borehole control the fracture width and stress intensity factor. The above two facts significantly simplify the calculations of aperture and stress intensity factor for a stable fracture propagating from an inclined borehole since the stress normal to the fracture plane and the stress perpendicular to the borehole are given by the general plane strain solution. Note that if am is not equal to a m , the =

A

Pressure sealing point Fig. 1. Boundary condition for an inclined well with a fracture~a general plane strain problem.

MORITA et

al.:

BOREHOLEBREAKDOWN--SEMI ANALYTICAL SOLUTION

55

J J

t

J

Fig. 2. Finite element mesh for a fractured borehole--2-D mesh (90° mesh) and 3-D inclined well mesh (180° mesh). out-of-plane stress also affects the stress intensity factor and fracture width. Let anl and tr m be in-plane, principal in situ stress given by trm = trH,

grin = COS2 7all + sin 2 )'try

(1)

Since out of plane stress does not affect the fracture width and stress intensity factor, we can simplify the problem as a two-dimensional plane strain borehole problem with a fracture induced along the principal stress direction trn2 as shown in Fig. 3. Fracture width and stress intensity factor are generated using a 2-D finite element model for the following conditions: (1) A borehole with a fracture, the fracture length being varied. (2) Well pressure Pw, hydrostatic pressure trm , stress difference trm - a m , and pressure within fracture Pf are independently applied. A semi-analytical solution is constructed for each boundary load. The semi-analytical solution is derived assuming a planar fracture without a borehole. The correction factors for the borehole are determined by fitting to the numerically generated fracture width and stress intensity factor. The finite element mesh was refined to give solutions with less than 1% difference from the solution given in Ref. [1]. Note that the fracture width is not given in Ref. [1]. However, since a standard finite element code gives a better accuracy for the fracture width calculation than the stress intensity factor calculation, the error of the fracture width should be also within 1%. Equations (TI.1)-(T1.6) are the semi-analytical solutions thus determined for the well pressure and fracture width when the stress intensity factor becomes equal to the fracture toughness. The borehole pressure is not applied to the fracture surface for these equations. Equations (T2.1)-(T2.8) are the semi-analytical solutions for the well pressure and fracture width when the stress intensity factor becomes equal to the fracture toughness. The borehole pressure for this case is applied to the part of the fracture surface (up to B); however, it is not applied to the rest of the fracture

surface. These solutions give less than 1% error for those values which can be checked from Ref. [1], although the error is not known for those values which cannot be checked with Ref. [1]. The fracture width and stress intensity factor under thermal stress or pore pressure are determined with the following two steps: (1) First, the reaction force to offset the thermal or pore pressure is determined analytically as shown in Fig. 4. (Appendix, Sections A. 1 and A.2 provide the derivations.) Assuming a plane fracture with no borehole, analytical solutions for the stress intensity factor and fracture width are derived. (2) A correction factor is multiplied for the borehole effect comparing with the results obtained by the finite element model. Note that the correction factor was determined only to the logarithmic function used for temperature and pore pressure distribution, hence, the correction factor cannot be applied to OH2

OH1

A

P0

A Fig. 3. Boundary conditions for the in-plane stress components.

M O R I T A et al.:

56 (A)

~

~

BOREHOLE BREAKDOWN--SEMI ANALYTICAL SOLUTION

mined to calculate the borehole breakdown pressure and fracture aperture including temperature, pore pressure and borehole inclination.

T(r)

PARAMETRIC ANALYSIS OF BOREHOLE BREAKDOWN PRESSURE

(B)

rD

Ect r ~ 2 f 1 rDTdrD_~.vct T T--V

a

Fig. 4. Equivalent fracture surface load to offset the temperature load: (A) borehole with a fracture under temperature field; and (B) equivalent fracture surface load.

general temperature or pore pressure distribution. Tables 3 and 4 show a set of equations thus deter-

The previous section explains the derivation of the well pressure, fracture length and fracture relation (Pw- A / R w - W relation) shown in Tables 3 and 4. Figure 5 shows example relations for a specific set of parameters. If the width Wmof the sealing point is given, the borehole breakdown pressure can be calculated as shown in the arrow. Note that Wm depends on the type of fracturing fluid (typically Wm for a short fracture is 0.254-0.381 mm (0.01-0.015 in.) for the bentonite base muds and 0.076-0.127mm (0.003-0.005in.) for hydraulic fracturing gels) and Wm may become large as dehydration progresses for some drilling fluids. Using the equations given in Tables 3 and 4, parametric analysis for borehole breakdown pressure was conducted. There are six factors which disturb the stress state around a borehole. They are:

Table 1. Basic solution for borehole breakdown

Borehole breakdown pressure (Tl.l)

(1) A ~ rw: P~ = - 3 a m + a m + K¢/[1.215v/-~ -- r,)] (2) 1.01r~ < A < 10r,, Pw = [K¢ -- an]( {n/2 + fl l%//l~ fl 2 - arcsin(fl)}F1 - (am - trm)[([33x/l -- [32 F 2 ] / [ ~ [ 3 ~

(T1.2)

F3]

(3) For a large A/r w, use F I = 1, F 2 = 1 and F3 = 1. For a small A/rw, use the following table for estimating F I , F2 and F3.

Equation of fracture aperture

(TI.3) (T1.4) (TI.5) (T1.6)

w = w, + w2 + w3 (1) For hydrostatic boundary stress: W I = 4 0 - v 2 ) A G 1/E x a m (2) For directional stress: W2 = 4(1 - v2)AG2/E x [am - am] (3) For borehole pressure: W3 = 8(1 - v:)rw/(rrE) x [G3 + ln(A/rw)]Pw (a) A ~ r,,: G l = 2 . 9 1 6 ( 1 - r w / A ),

G 2 = 1 . 4 5 8 ( 1 - r w / A ),

G 3 = 2.29(A/rw)-ln(A/rw)

(b) 1.01r W< A < 10rw:Gl, G2, G3 given the following table. (c) A larger than 10rw:G1 = 1, G 2 = 0 , and G3 = 0 . 3 9

I 1 2 3 4 5 6 7 8 9 10 1! 12 13 14 15 16 17 18 19 20 21

A/R~ 1.01 1.02 1.04 1.06 1.08 1.I0 1.15 1.20 1.25 1.30 1.40 1.50 1.60 1.80 2.00 2.50 3.00 4.00 6.00 8.00 10.00

F1

F2

F3

1.228 1.227 1.212 1.196 1.183 1.171 1.146 1.125 1.109 1.095 1.075 1.060 1.049 1.035 1.026 1.015 1.009 1.004 1.002 1.001 1.000

1.236 1.232 1.222 1.210 1.201 1.192 1.173 1.157 1.145 1.134 1.117 1.105 1.095 1.082 1.072 1.063 !.053 1.042 1.034 1.025 1.018

1.229 1.224 1.214 1.200 1.200 1.175 1.152 1.133 1.117 1.104 1.084 1.069 1.059 1.044 1.034 1.024 1.015 1.013 1.010 1.008 1.005

G1

G2

G3

0.231

0.107

0.0994

0.385

0.160

0.162

0.495

0.189

0.206

0.638

0.207

0.258

0.814

0.197

0.324

0.926 0.964 0.991 1.000 1.000

0.147 0.114 0.077 0.058 0.047

0.360 0.372 0.383 0.385 0.385

ID Oh2

~h I

J

r~

M O R I T A et al.:

BOREHOLE BREAKDOWN--SEMI

57

ANALYTICAL SOLUTION

Table 2. Basic solution for borehole breakdown from an open crack Borehole breakdown pressure from an open crack S = ~[arcsin(B/A)-- arcsin(fl)],

~ = 2 Ax~x//A~, fl = r , / A ,

S1 = ~flx/1 - f12 F3

(T2.1)

(I) A ~ r w e , = [K c + { -3or m + a m ) x 1.1215 n , , / ~

- rw)l/[l.1215.,/~(A - rw) + S]

(T2.2)

(2) 1.01rw < A < 10r,~

Pw = [jc _ an]( {rr/2 + f l x / l - f12 _ arcsin(fl)}rl - (trm - an2)(fl3x/1 - f12 F2]/[S + S I ]

(T2.3)

F1, F2 and F3 are given in Table 1. (3) F o r a large A/r~:(A/r,..GT.3), use F I = 1, F2 = 1 and F3 = 1.

Equation o f fracture aperture W= W~+W 2+W 3+W4

(T2.4)

(1) F o r hydrostatic b o u n d a r y stress W~ = 4(1 - vZ)AH lIE x {l x / f ~ 5q- 2 + ( 2 / n ) ( / t x / l - / t 2 - arcsin/t)ln[(l + x/1 - q2/q]} x au~

(T2.5)

(2) F o r directional stress W2 = 4(1 -- v2)AH2/E x [trm -- am][(2/n)fl31,,/i~i-~2x/l -- q2/q2]

(T2.6)

(3) F o r borehole pressure W 3 = 8(1 - vZ)AH3/(nE) x P w f l l ~ / ~ 2 - ~ 21n[(l + 1,,/~-q2/rll

(T2.7)

(4) For fracture surface pressure W4= 8(1 -v2)AH4/(nE) x P~ x {r/arcsin(,/)~/1 --t/2/q -r/In(r/)- arcsin(fl)ln[(l + 1 , , / ~ q2)/*/]} (T2.8) where H1 = 1, H2 = I, H3 = 1, H4 = 1 are good approximations for B/> 1.5rw. For B ~<1.5rW, they are given by a function of A / B and B/r,~. For example H I - H 4 are given in the following table for B/rw = 1.333. I

A/B

H1

H2

H3

H4

1

1.1

1.049

I. 116

1.035

1.079

2

1.3

1.047

1.076

0.998

1.073

3

1.5

1.030

1.057

0.983

1.062

4

2.0

1.020

0.978

0.950

1.025

5

2.5

1.013

0.945

0.950

1.006

6

3.0

1.011

0.928

0.950

0.995

7

4.0

1.010

0.914

0.950

0.998

8

6.0

1.009

0.892

0.950

0.993

9

8.0

1.005

0.868

0.950

0.983

10

10.0

1.003

0.849

0.950

0.975

(A) irregularity of borehole shape (B) temperature cooling (C) inclined well (D) pore pressure build-up (E) nonlinear rock deformation (F) shale swelling. Since shale swelling normally increases lost circulation pressure by increasing the stress state around a borehole, only the effect of (A)-(E) are discussed. Borehole shape may be elliptical, but the most irregular form of borehole shape is a borehole with a notch (the notch represents a natural fracture or a vuggy pore). Hence, the irregularity of borehole shape is treated in this work as a fracture propagation from an existing fracture. Figure 6 shows the prediction of the theory on the typical fracture extension pressure from a borehole. Note that if a wide fracture exists around a borehole, the fracture extension pressure should start from a certain fracture length B. If the fracture toughness is large, there may initially be a small decline in the fracture extension pressure as shown in Fig. 6, although most sandstones

D Oh2

ohl

j/W

B DPI A

do not have a large fracture toughness. As the fracture enlarges, the fracture extension pressure increases. Fracture extension pressure normally starts declining after a peak pressure if the fracture aperture becomes wide enough to allow drilling fluid entry (Note: B = rw for no drilling fluid entry into fracture). Hence, for the normal case, the fracture extension pressure which allows drilling fluid to invade into a fracture coincides with borehole breakdown pressure. However, if stress disturbance is significant, borehole breakdown does not occur due to further stable fracture extension even after drilling fluid entry. As shown in Fig. 6, the peak fracture extension pressure is lowered in accordance with the magnitude of stress disturbance around a borehole. The discussion of factors affecting the peak fracture extension pressure (borehole breakdown pressure) follows. In situ s t r e s s

Figure 7 shows the in situ stress effect. It is a primary factor affecting borehole breakdown pressure. The new

58

MORITA et al.:

BOREHOLE BREAKDOWN--SEMI ANALYTICAL SOLUTION

Table 3(a). Breakdown pressure with pore pressure and temperature [Note: applicable only for q >Po~/ln(A/rw), qt < T~/ln(A/r,)] Pore pressure distribution P = Po~ - q(tD)ln(r/rw)

P =0

for r < r~ exp(Pow/q) for r > r wexp(Pow/q )

(T3.1)

Temperature distribution T = Tw--qvln(r/r~)

T=0

for r 1rwexp(Tw/qv )

%=(1--Ci/Cb)(1--2v)/E,

~,=2/A/r~,

(T3.2)

fl=r~/A

I~ ~h2

~HI ~ (TH

am = cos2 yau + sin 27av (~hl

Borehole breakdown pressure

(1) A ~ r~ P~ - Pr - 3am + am + E(aT~ + %Pow)/(1 -- v)

+ KC/[l.1215x/~ -- r~)]

(T3.3)

(2) 1.01rw < A < 10rw Pw = Pf+ E(%q + ctqr)/[2(l - v)] +[K ¢ -- {am] + Pf+ E(aqz - 2ct T w + aoq

I

-2aoPo~)/[4(l - v)] x ~{n/2 + fix/12 - fl: arcsin(fl)}F1

[

-- (am -- am)(fl3~/1 -- f12F2 - E( ~qT + ~0q)/[4( 1 -- v)]

rw A

x ({2 arcsin(fl)ln(fl) - 7r ln(fl) + 2 arcsin(fl) --2 arcsin(fl)ln[arcsin(fl)] + [arcsin(fl)]3/9 + [n ln(n/2) - ~ - n 3/72]}]/[~flx/1 - f12F3]

(T3.4)

(3) For a large A/rw, use F1 = 1, F 2 = 1 and F3 = 1. For a small A/r w, use Table 1 for estimating F1, F2 and F3. Table 3(b). Fracture aperture with pore pressure and temperature [Note: applicable only for q >Pow/ln(A/rw), qt < Tw/ln(A/rw)] Equation o f fracture aperture

(T3.5)

w = w , + w2 + w~ + w :

(1) For hydrostatic boundary stress Wl = 4(1 -- v2)AG 1/E × {a m + Pf + E(~tqT -- 2ctTw+ % q -- 2% P0~)/[4( I -- v)]}

(T3.6)

(2) For directional stress W2 = 4(1 - v2)AG2/E × [am - am] (3) For borehole pressure W3 = 8(1 - v2)r,/(uE) x [G3 + ln(A/rw)]{P w - P f - E[ctqx + %q)/(2(1 - v)]}

(T3.8)

WT = 2(1 + V)(~qTq- %q)Al2/n" --t.1{s/(s 2 -- fl')**0.5} {(n/2)ln(s) -- arcsin(fl/s)ln(s) + 0.5[n ln(u/2) -- u -- n 3/72] -- arcsin(fl/s )ln[arcsin(fl/s)] + arcsin(fl/s) + [arcsin(fl/s)]~/18} ds +ln(fl){,/1 - f12 + (2/,0fl ln(fl) - (2/.) l ~ / V S ~~: -

f r a c t u r e t h e o r y p r e d i c t s significantly h i g h b o r e h o l e stability at h i g h in s i t u stress if t h e b o r e h o l e size is small. Borehole size

F i g u r e 7 s h o w s t h e b o r e h o l e size effect. A slim h o l e m a y be several h u n d r e d psi s t r o n g e r f o r lost c i r c u l a t i o n p r e s s u r e t h a n a l a r g e hole. T h i s size effect p r e d i c t e d f r o m t h e n e w t h e o r y m a t c h e s the field o b s e r v a t i o n , w h i l e t h e conventional continuum theory predicts a constant boreh o l e b r e a k d o w n p r e s s u r e r e g a r d l e s s o f the b o r e h o l e size. Modulus

effect

F i g u r e 8 s h o w s t h e m o d u l u s effect. N o t e t h a t f r a c t u r e w i d t h , Wm, w h i c h a l l o w s d r i l l i n g fluid e n t r y is r e a s o n -

arcsin(fl)}[ /

(T3.9)

a b l y c o n s t a n t (0.254-0.381 m m o r 0 . 0 1 - 0 . 0 1 5 i n . ) , acc o r d i n g to the l a b o r a t o r y results, b e f o r e the w i d t h Wm grows with fracture extension. Hence, formation Y o u n g ' s m o d u l u s p r i m a r i l y affects the f r a c t u r e i n i t i a t i o n pressure. N o t e t h a t the c o n v e n t i o n a l l i n e a r c o n t i n u u m t h e o r y d o e s n o t i n c l u d e Y o u n g ' s m o d u l u s in the e q u a t i o n o f t h e b o r e h o l e b r e a k d o w n pressure. Effect of width of pre-existing fracture

I f a c r a c k w i t h w i d t h , W0, exists a r o u n d a b o r e h o l e , the p r e - e x i s t i n g f r a c t u r e w i d t h s h o u l d be s u b t r a c t e d w h e n u s i n g Fig. 8. F o r i n s t a n c e , s u p p o s e the w i d t h a l l o w i n g d r i l l i n g fluid e n t r y i n t o f r a c t u r e is 0.254 m m (0.01 in.), the p r e f r a c t u r e w i d t h is 0.229 m m (0.009 in.),

et al.:

MORITA

BOREHOLE

BREAKDOWN--SEMI

and E l ( 1 - v 2) = 6890 MPa (1.E6 psi), then a curve for W m E / ( 1 - v 2) = (0.254-0.229) x 6890 = 172 m m - MPa or (0.01 - 0 . 0 0 9 ) × 1. E6 = 1. E3 in.psi should be used. The result shows that preexisting fracture width is an important factor for breakdown pressure. The derivation of P ~ - A / R w - W relation for a borehole with a pre-existing crack is shown in the Appendix, Section A.3.

Nonlinear deformation of rock There are three non-linear deformations often observed for sedimentary rocks. They are Mode I nonlinT a b l e 4(a). B r e a k d o w n

ANALYTICAL

SOLUTION

59

earity (crack closure or Mode I microcrack growth), shear type plasticity, and plasticity due to pore volume crushing. The Mode I nonlinearity has a similar effect as macrocrack growth, hence, it can be treated approximately as a part of a macrocrack growth problem. Plastic nonlinearity occurs if a rock is soft relative to in s i t u stress. Figure 9 shows the effect of plasticity for a relatively weak rock (the stress-strain relation is given in the figure). The tangential stress may become compressive due to plasticity if borehole pressure is too high. For some stress combinations, tangential stress remains

p r e s s u r e f r o m a n o p e n c r a c k w i t h p o r e p r e s s u r e and temperature [Note: a p p l i c a b l e o n l y for

q > Po~/ln(A/rw), qt < Tw/ln(A/r~)] Pore pressure distribution e = Po~ - q(tt))ln(r/rw) P = 0

for r < r w e x p ( P 0 ~ / q ) for r > r~ exp(Pow/q)

(T4.1)

Temperature distribution T = T~--qTln(r/rw)

for r < r w e x p ( T w / q x ) for r >1 r~ exp(T~/qT )

T = 0

(T4.2)

~0 = (1 -- Ci/Cb)(1 -- 2v)/E, (=2

Ax/A~,

fl=r,~/A,

D ~h2 SI=~//

lxfi~---//2F3

S = ~ [ a r c s i n ( B / A ) - arcsin(//)],

Borehole breakdown pressure f r o m an open crack

P~= --Pf+[KC+{--3trm+trm+E(~Tw+%Pow)/(1 ×l.1215x~(A~w)]/[1.1215x/~-r,.)+S]

-- v)}

(

(T4.3)

w

~

"i

I

~Pf

(2) 1.01rw < A < 10r w P~ = Pf + [S1 × E ( % q + ctq-r)/{2(1 - v)}

_

+ K c - {tr,u I + Pf + E(etqx - 2ctT, + % P 0 , ) / [ 4 ( 1 - v)]} × ~"{ ~ / 2 + f l ~ - - - - / / 2

-

B

_ a r c s i n ( f l ) } r l - (tr m - a m)~fl31xf~--fl2r2

--E(ctqx + % q ) / [ 4 ( I -- v)] × ~{2 arcsin(fl)ln(fl) -- ~ In(//) + 2 arcsin(//)

-I

--2 arcsin(//)ln[arcsin(//)] + [arcsin(fl )]3/9 + ( ~ l n 0 z / 2 ) -- r~ -- n3/72)}]/[S + S1]

(T4.4)

(3) F o r a l a r g e A/rw, use F I = 1, F 2 = 1 a n d F 3 = 1. F o r a s m a l l A/rw, use T a b l e 1 for e s t i m a t i n g F I , F 2 a n d F3.

Table

4b. F r a c t u r e

aperture

at an

c r a c k w i t h p o r e p r e s s u r e and temperature [Note: a p p l i c a b l e q > Pow/ln(A/rw), q, < Tw/ln(A/rw)]

open

only

for

Equation o f fracture aperture W = w, +

w2 + W3+ wpT

(1) F o r h y d r o s t a t i c b o u n d a r y

(T4.5)

stress

W~ = 4(1 - v2)AH 1/E × [x/1 - r/2 + ( 2 / n ) ( B x / l

- f12 _ a r c s i n fl)ln((1 + x / 1 - r/2)/r/)]

× [o m + pf + E(CtqT -- 2ct T w + % q -- 2% P0w)/(4( 1 - v ))]

(T4.6)

(2) F o r directional stress W 2 = 4(I -- v Z ) A H 2 / E × [trm -- trml[(2/n)fl31~--fl2--fl2x/1 -- rl2/q2 l

(T4.7)

(3) F o r b o r e h o l e p r e s s u r e W 3 = 8(1 - v2)AH3/(rcE) × [Pw - P f - E(~tqT + % q ) / ( 2 ( 1 -- v))] × f l x / l -- f12 l n ( ( l + x / 1 -- tl2)/q)

(T4.8)

(4) F o r f r a c t u r e s u r f a c e p r e s s u r e W4 = 8(1 - v2)AH4/(nE) x [Pw - Pr] × [ r / a r c s i n ( r / ) x / 1 - ~/2/r/ - r / l n ( q ) - arcsin(//)ln((1 + x / 1 - r/2)#/)]

(T4.9)

W~T = 2(1 + v)(~qz + % q ) A lr/ arccos(r/) -- w / 1 - , 2 _ l n ( 2 ) x / ~ _ r/z + ( 2 / n ) / / ( 1 - ln(fl))ln((1 + x / 1 - q2/q) _(x/l

_ n2 _ ( Z / n ) / / I n ( ( 1 + x / 1 - n2/n))ln(//)]

where H 1 to H 4 are c a l c u l a t e d as shown in T a b l e 2.

(T4.10)

60

MORITA et al.:

BOREHOLE B R E A K D O W N - - S E M I ANALYTICAL SOLUTION /

48.2 41.3

/

/-" /.." /." /'"

55.1

Kc -- 59 MPa.cm It2 . - / . ' /

/

/

'

"

~

J

_

Ko-- 3 0 ~ , " / / ~ o

/'./

34.5 eL

27.6 -

" " ,,-:,, _/~ /

_

eL

20.7

/ /

59 ~ - I " _...~--...i.--~.-

p-..::...:~._._..----_.7.2.-"'-o"•-'""......--•"" 2.~ ....~.'2.'2"-''

"/

13.8 6.9 0

59

~ - - - - 3 -

_ __1.4

5

\ ."x,~"~ ~ ~ -. \ ~ 0.875

_

\.

6

-,

....

"~,

9

to

-5~ . . . . . . .

"~.

k"-~,,,~ - - .

,

.,,'..,\~ -. \

% > I

80

-

o

g-,

7

~..,

N.~ ~

-..,,

"~59 ..

59 ".. " . , \ \ \ 1.750 _

", ~,.

"-.~

..

"x.

"'-.

\ \ \ 0

",..,,.\\'~.3o

--.,,

--.

~.\

\\\

-,,

pa~)" N\ "\ 59~~

2.625 ~'~"

(~- I ~ P

O/./2= 1.72MPa)

(OH1 = 0 MPa, ott 2 = 0 MPa)

X

"~, "'.. N.

"~'~.

\

Fig. 5. Effect of in situ relation on P~ - A / R w -- W relation (R~ = 6 in.).

compressive without opening a fracture. Such plastic effects increase borehole breakdown pressure. Table 5 summarizes the well pressure inducing fluid invasion into the fracture. A failure limit is imposed to plastic strain to those cases where tangential stress never becomes tension, assuming nonuniform deformation destabilizes the borehole if plastic strain exceeds a certain value. The table compares the nonlinear fracture theory with the conventional linear elasticity theory. The

nonlinear theory consistently predicts a higher fluid invasion pressure into the fracture.

Irregular borehole shape The most serious form of irregular borehole shape is a wide crack around a borehole. Figures 10 and 11 show the effect of a wide crack upon breakdown pressure. The borehole breakdown pressure is significantly reduced by wide cracks, but even with a relatively large crack,

Table 5. Wellbore pressure inducing fluid invasion into fracture 5(a) Rock type: weak (unconfined compressive strength = 150psi) (O'H~ GV)

Well angle 0° 90 °

( - 0 . 0 5 kpsi, - I kpsi) Linear Nonlinear continuum fracture (psi) (psi) 1000 500

1060 630

5(b) Rock type: intermediately weak (unconfined compressive strengths = 1 kpsi) ( - 0 . 5 kpsi, - 1 kpsi) Linear Nonlinear continuum fracture (psi) (psi) Well angle 0° 45 ° 90 °

I000 750 500

1600 1230 840

( - 0 . 5 kpsi, - 2 kpsi) Linear Nonlinear continuum fracture (psi) (psi)

( - 1 kpsi, - 2 kpsi) Linear Nonlinear continuum fracture (psi) (psi)

1000

1900

2000

2820

- 500

500

1000

1650

M O R I T A et al.:

~

BOREHOLE BREAKDOWN--SEMI

ANALYTICAL

61

SOLUTION

B = Rw

II /

Drilling fluid entry into fracture

~

B

No stress disturbance

~

I I \ . / P o r e pressure

~

I ~

[

~

lid-up

¢0

\ I I

Solid free fluid (no permeability is assumed)

~~ ~ ~ ~ ~ ~ ~ ~ ~

1 ,R w

2,R w

3,R w A

Fig. 6. Typical fracture extension pressure for various stress disturbances.

borehole breakdown is still significantly higher than the in situ stress if in situ stress is high. Thermal

cooling effect

fracture, the temperature effect is offset when a crack size is very small.

Effect of borehole

Because the thermal expansion coefficient is relatively large, borehole cooling lowers borehole breakdown pressure. The borehole breakdown pressure reduces with time after a formation is exposed to a cool drilling fluid. Figures 12 and 13 show the effect of thermal cooling. The effect is high for a tight rock, but since a tight rock has a large fracture propagation resistance for a small

inclination

Borehole inclination enhances the nonuniform stress field around a borehole, but the nonuniform stress field quickly diminishes with distance. Hence stable fracture theory predicts less sensitivity to borehole angle. Figure 14 shows the effect of borehole inclination. The well inclination effect is non-trivial for a formation with a low Young's modulus. However, once a stable crack starts

41.3

4

~

2

3 cm.MPa

34.5 34.5

\

\ \

"~ k

(OHI=5"17MPa'OH2 =6'89MPa)

~(OHI=5.17MPa,

OH2=5.17MPa)

27.6 7E3

e 21).7

e 217.7 .44E3

13. 8

13.8 (°ttl

1

6.9 -

0

I

I

2.5

5.1

7 2 M P a ~ (oN! = 1.72 MPa, olt 2 = 3.45 MPa)

I

I

I

I

I

7.6 10.2 12.7 15.2 17.8 R w (cm)

Fig.

6.9

7. Borehole breakdown pressure vs Rw and in situ stress [ W E / ( l - v 2) = 25000 in.*psi, K c = 600 psi*in J/2].

0

Linear continuum theory (o t = 1 MPa)

I

I

2.5

5.1

I

[

I

I

I

7.6 10.2 12.7 15.2 17.8 R w (cm)

Fig. 8. Borehole breakdown pressure vs Rw and WE~(1 - v 2) (in situ stress -- - 5 0 0 psi, K c = 600 psi*inJa).

MORITA et al.: BOREHOLEBREAKDOWN--SEMI ANALYTICAL SOLUTION

62

-3.44 m -2.76 --

t~H = -3.4 MPa \

'\" q°v=-6'9--'--~MPa

\ ,~ ~-2.07

~

~ ~

'

i5s~i!!!!!!! !

oo

-

'\

~

-0.04 Str~.s~4st0r.~i8n0r.;12ti0°160.20

~

i

110.0 ~ i i [ [ 96.5 ~- ~Radial strain f x

table fracture

!

entry into f r a c t u r e T /

16

8628"

-1.38= t~

-0.69 --

XXXx\ "'\'\. ~ \\\ "\.

~

/" / 7" /

~ l .~ 41.3 [ - - /

•

Pc = 41.3 MPa Pc=13.SMPa

_ 6

:g~

-- 4

< "\\Oi._.x\.\t...__.~J

0 m

"< 27"6 ~ 13.8 ~-I

o= 0.69 1.72

I 3.45

I 6.89

I 10.3

I 13.8

~ ~

-- 2

~Pc =0

0I ~ t I I L 0 -0.04 0 0.04 0.08 0.12 0.16 0.20

Borehole pressure (MPa) Strain Fig. 9. Nonlinear effect on tangential stress (evaluated at the borehole top surface, intermediately weak sandstone).

extending, borehole angle effect is quickly reduced (Figures 15 and 16).

Pore pressure build-up effect Pore pressure build-up occurs if formation permeability becomes small compared with drilling fluid cake permeability. Pore pressure build-up increases total stress, which results in preventing fracture propagation. However, if a tight formation allows pressure build-up at the fracture tip, it offsets the higher in situ stress and reduces fracture propagation pressure. Figure 17 shows the pore pressure build-up effect for a formation with a relatively small permeability (not extremely small though). Fracture extension pressure is reduced by the higher pore pressure when the fracture size is small. However, the higher total stress offsets the reduced tip pressure, which results in higher fracture propagation pressure. It is emphasized that pressure build-up at the

fracture tip significantly reduces borehole breakdown pressure and fracture extension pressure for an impermeable formation since the pressure at the fracture tip is always equal to borehole pressure when a formation has very low permeability. Dehydration effect

Dehydrated mud plugs fracture tip with a wider fracture width, Win, to prevent fracture pressure from transmitting to fracture tip. As fracture length becomes larger, the width, Wm, increases from 0.38 mm to a larger width. Figures 8 and 10 show that a wider width gives a larger fracture extension pressure, especially for a small fracture. Fracture reopening

A drilling fluid which acts as a perfect sealant after dehydration does not leak into a pre-existing fracture if

41.3

41.3

34.5

34.5

27.6

27.6

~

~,~ 20.7

~ 20.7

13.8

13.8

MPa

6.9

6.9

0 22E3 0.11E3 ~ ,' 0.44E3 I I I 0 25 51 76 B (cm)

~

I 102

I 127

I 152

Fig. 10. Borehole breakdown pressure for a well with a wide crack (in situ stress = - 500 psi, K¢ = 600 psi*in.~/2, Rw = 6 in.).

0

L 25

I 51

-3.45 MPa -0.69 MPa I I 76 102 B (cm)

I 127

I 152

Fig. 1I. Borehole breakdown pressure for a well with a wide crack [WE~(1 - v2) = 25000 in.*psi, Kc = 600 psi*in, t/2, RW= 6 in.].

MORITA

et al.:

63

BOREHOLE BREAKDOWN--SEMI ANALYTICAL SOLUTION Fluid front position (in.)

19.3

10 I

0

20 I

30 I

40 [

50 I

2800

2400

16.5 -After 0 ~ t e m p e r a t u r e

eL --t

60

2000

13.8

O

e~ O o~

1600 ~

11.0

.---'-

. ~ . ~-" " ~

8.27

-----"

After2.3 days

--'"" Horizontal in I 25.4

5.51

1200 ""-

I 50.8

situ

After 11.6 days

stress

[ 76.2

I 127.0

101.6

800 152.4

Fluid front position (cm) Fig. 12. T r a n s i e n t t e m p e r a t u r e effect ( 1 0 0 ° F , a n = - 1000 psi, K t = 0.0116 cm2/sec, R w = 6 in., K c = 0, E = 2 E 6 psi, • = 0 . 8 E - 5 p e r OF, v = 0 . 1 , W m = 0 . 0 1 in.).

w e l l b o r e p r e s s u r e is less t h a n the well pressure g i v i n g zero h o o p stress or the in situ stress, w h i c h e v e r is larger. O n the o t h e r h a n d , a drilling fluid w h i c h acts as a leaky s e a l a n t (i.e. a s e a l a n t w h i c h c a n g r a d u a l l y b u i l d u p

pressure in a fracture) leaks i n t o a fracture at a p r e s s u r e less t h a n the m i n i m u m in situ stress b u t the leak stops if the wellbore p r e s s u r e is less t h a n the in situ stress. H e n c e the f o l l o w i n g e q u a t i o n s gives the m o s t pessimistic

Fluid front position (in.) 34.5

eL

0

12.5 I

25.0 I

37.5 I

50.0 I

62.5 I

75.0 [

87.5 I

100 5000

-- 4000

27.6 -oling, E = 5.5E7 MPa

-- 3000 .~

.~ 20.7 ~

50 degree cooling, E = 5.5E7 MPa

9 ~

".. ",.

13.8

" ~ / ~

. . . . . . . . . E = 1.4E7 MPa ..... E=O.34E7MPa

2000

" - ~ 2 -.7._ : . : _ __. . . . . . . . . . . . . . . . . . . . . . . . . 6.89

0

[ 32

I 64

I 95

I 127

I 159

[ 191

I 222

1000 254

Fluid front position (cm) Fig. 13. Temperature effect for various Young's modulus (50°, gn = - I 0 0 0 psi, K, = 0.0116 cm:/sec, t = 2.3 days, Rw = 6 in. Kc=O, • = 0.8E-5 per °F, v =0.1, Wm= 0.01 in.).

64

M O R I T A et al.: 82.7

0

BOREHOLE BREAKDOWN--SEMI ANALYTICAL SOLUTION

15

30

45

60 I

75 I

90

12

WmE/(l-v 2) = 1.75 x 103 cm.MPa

1.31 x 103 /

68.9

~ --

10

g" 8.75 x 1

gh

55.1

_

0

2

/

~

4.38 x 102

8

.'Z'.

.o

..s :::::::::::::~ii~:~'~2~-.):!!.xlO2/8.75xlO ~

e~

..................

o m

.o

41.3

...............

O "O

........... ::::::::::::::::::::::::::::::::::::::::::::--27.6

o O

"-------___________

.o

o m 13.8

--

I 25.4

I 50.8

I 76.2

I 101.6

I 127.0

2

0 152.4

Well angle (o) Fig. 14. Elastic modulus effect on borehole breakdown pressure (a n = - 2 5 0 0 p s i , K ~ = 0).

a, = - 5 0 0 0 p s i ,

Rw= 6in., B/Rw = 1,

Fluid front position (in.) 8.27

0

I0 I

20 I

30 I

40 I

50 I

60 I

70 I

80

90 I

100 1200

\ 7.58

--

1100

-

1000

OH = - .4MPa ~" eL 6.89

e~ o v = -6.9 MPa

.o

-

9oo

5.51

-

800

4.82

-

700

6.20

~

ca,

t =

~

l

l

F r o m t o p a = 0, 15, 30, 45, 60, 75 and 90 degree

.o

~

4.13

~

-I! 600

3.45 0

1 25.4

I 50.8

t 76.2

I 101.6

I 127.0

I 152.4

I 178

I 203

I 229

500 254

Fluid front position (cm) Fig. 15. Borehole breakdown pressure for a low in situ stress formation [intermediate modulus rock: W E / ( I - v 2 ) = 25000 in.*psi, Rw = 6 in.].

M O R I T A et al.:

BOREHOLE BREAKDOWN--SEMI ANALYTICAL SOLUTION

fracture aperture became approx. 0.1 mm (0.004in.). For fluids which acted as leaky sealants, fracture reopening did occur at a pressure less than the minimun in situ stress but the leak stopped with the fracture inflation. The fracture inflation occurred until the fracture pressure became high enough to restart fracture propagation.

fluid loss pressure into a preexisting fracture previously invaded by a drilling fluid. For a drilling fluid with perfect sealing capability: Pw = Max{I trm l, I - 30"HI+

0"/'/2 - -

e0l}.

(2)

For a drilling fluid without perfect sealing capability: Pw = IOnl ]

(3)

CALCULATION PROCEDURE AND EXAMPLE CALCULATIONS

The experiments in Part I of this paper showed that the dehydrated mud embedded between fracture surfaces should be sufficiently thick and strong to act as a perfect sealant. The test samples which were split apart showed that the 1.92 g/cc (16 ppg) water-base mud and LVT oil base mud formed a mud cake thick enough to cover the rough fracture surface. Some mud cake squeezed out into the borehole from the fracture indicated that it can seal fractures even if larger particles are embedded between surfaces and widen the preexisting fracture aperture. On the other hand, the 1.92g/cc (16ppg) mineral and diesel oil base muds and all the 1.2 g/cc (10 ppg) muds could not seal a fracture gap which had been partially induced by grain mismatching at fracture surface and embedded hard components contained in drilling solids. Equations (2) and (3) give the most pessismistic fracture reopening pressures for drilling fluids with and without sealing capability, respectively. However, if taking the average value of the fracture reopening pressures, the experiments showed that for muds with perfect sealing capability, fracture reopening did not occur until 0

15 I

20.7

65

The equations of well pressure and fracture width shown in Tables 3 and 4 (Table 3 for B = r~ and Table 4 for B/rw.GT.1.33) give P ~ - A / r w - W m E / ( I - v 2) relations as shown in Fig. 5 for various temperature, pore pressure and pre-crack sizes. If the critical fracture aperture Wm [or W m E / ( I - v2)] with which a drilling fluid can invade into a fracture is given, the borehole breakdown pressure can be calculated. Since the fracture width is given as a function ofA/rw, a numerical iterative method is required to find A/rw from Wm if the calculation is carried out with a computer program instead of the graphical method shown in Fig. 5. Inclined wells and thermally cooled wells have a significantly low tangential stress around a borehole. A very stable fracture grows without a lost circulation. For these stable fracture growth problems, the equations for notched borehole shown in Table 4 must be checked even if the borehole surface is initially smooth. The maximum pressure found by varying B (the mud invaded crack radius) is the lost circulation pressure. The equations predict the borehole breakdown pressure accurately if the correct input data

30 I

45 I

60 I

75 [

90

3000

(OH, OV, Kc, B/Rw) = (-3.4 MPa, -6.9 MPa, 6.6 MPa.cm I/2, 1) .................... 17.2 --

-1. . . . . . . . . . . . . ". . . . . . "

/ ~'

"

4

/

(-3.4, -6.9, O, 1) ~ ...............

'

"'''"

" ' " ' ' " "

'''"

2500

"'"

...............

(-3.4, -10.3, 6.6, 1) 13.8--

~--~ I Rw i

-- 2000

I I I,LI

(-3.4, / -6.9, 6.6, 1.333)

o

(-3.4, -6.9, 0, 1.333) m

o

(-3.4, -10.3, 6.6, 1.333)

10.3

t (-3.4, -10.3, 0, 1.333)

-- 1500 Q

g t.,

/l

6.89 ~ Conventional theory

.~ (-3.4, -6.9, 6.6, 5) (-3.4, -6.9, 0, 5)

(-3.4, -10.3, 6.6, 5) (-3.4, -10.3, 0, 5)

~

g -- 1000

(-3.4, -6.9, 0, 1)

/

,, " " ' ~

500

3.45 ~ " ~ . ~ "(•" "

0 0

.

-

3 . 4 , -10.3, 0, 1)

I

I

I

I

15

30

45

60

I ""~-~.. 75

0 90

Well angle (°) Fig. 16. Fracture extension pressure for a rock with a low Young's modulus (Rw = 6in., K c = 0, E = 0.2E6 psi, v = 0.1, Wm= 0.01 in.).

66

et al.:

MORITA

BOREHOLE

BREAKDOWN--SEMI

ANALYTICAL

SOLUTION

Fluid front position (in.) 0

12.5

25.0

37.5

50.0

62.5

75.0

I

I

1

[

I

I

34.5

87.5 I

100 5000

4000

27.6 --

o

X~,

---

E=5.5E7MPa

......

E = 1.4E7 M P a

ga. O

XX \\

20.7 --

\ x

3000

.o

E = 0 . 3 4 E 7 MPa

o O

.o tL 2000

13.8 --

~ ~ . ~ _ . :

6.89

porepressure ...............

---~.~-__ --------------

I

I

I

I

I

I

I

32

64

95

127

159

191

222

1ooo 254

Fluid front position (cm) Fig. 17. P o r e p r e s s u r e b u i l d - u p effect o n f r a c t u r e e x t e n s i o n p r e s s u r e ( a n = - 2 5 0 0

psi, Td = I 0 0 , Rw = 6 in., K c = 0, v = 0.1,

% = 0.7, Wm= 0.01 in.). are given. One problem of these equations is that they do not take into account the nonlinear behavior of rock. Generally the nonlinear behavior is negligible during the tensile failure, however, as explained for the weak rock behavior, the nonlinearity may affect the borehole breakdown significantly even during a tensile failure if the porosity of rock exceeds 30%. For solid free drilling fluids, the stable fracture growth is dominantly affected by the fluid pressure distribution within the fracture. These equations give only a crude approximation for these special problems. The following input data are required to calculate the borehole breakdown pressure: (a) Well radius rw and well inclination. (b) The radius B:rw plus the surface crack length. (c) In situ stress: vertical and horizontal (negative sign for compression). (d) Young's modulus, Poison's ratio, rock grain compressibility: Note that the Young's modulus to open a fracture is about half of the Young's modulus measured using a cylindrical sample due to the nonlinearity to open a fracture. Hence if the Young's modulus is measured by opening a fracture, the measured value is directly used as an input parameter; but if it is measured with a cylindrical sample, the Young's modulus must be halved for input data. (e) Pore pressure at the borehole surface and pressure gradient: these values must be calculated

at time t using the equation given in the Appendix (Section A.2). (f) Temperature at the borehole surface and temperature gradient: these values must be calculated at time t using the equation given in the Appendix (Section A. 1). (g) The critical fracture width Wm which allows mud fluid invasion into the fracture tip: Wm is between 0.254 and 0.381 mm (0.01 to 0.015 in.) for standard drilling fluid. Wmremains approximately constant if the API filtrate loss is less than 0.5 co. However for high fluid loss mud, Wm grows with fracture length. For example Wm vs Rr for the 1.92g/cc (16ppg) water-base mud used in this work is given by Wm= 0.254 + 0.051 X Rr (mm) or 0.01 + 0.002 x Rf (in.). For a stable fracture growth problem (inclined well or thermally cooled well), Wm must be increased with the fracture length. Table 6 shows the example calculations using realistic borehole data for a flawless borehole. Since the borehole is assumed to be flawless, the prediction matches the borehole breakdown pressure during a casing shoe test. Two borehole breakdown pressures are calculated in the table. PBW denotes the pressure initiating drilling fluid invasion into a fracture and PB denotes the pressure inducing lost circulation. Note that a borehole breakdown induces a fracture, but it may stop propagating if it is stable.

MORITA

et al.:

BOREHOLE

BREAKDOWN--SEMI

ANALYTICAL

67

SOLUTION

T a b l e 6. E x a m p l e calculations o f b o r e h o l e b r e a k d o w n pressure or. (psi) - 500 - 500 - 500 - 500 - 500 -500 - 500 - 500 -500 - 500 - 500 - 500 - 500 - 500 -500 - 500 -500 - 500 -500 -500 -500 -500 -500 -500 - 500 -500 -500 - 500 -500 -500 - 1000 - 1000 - 1000 - 1000 - 1000 - 1000 - 1000 - 1000 - 1000

av (psi)

Well angle (°)

- 1000 - 1000 -1000 - 1000 - 1000 - 1000 - 1000 - 1000 - 1000 - 1000 - 1000 -1000 -1000 - 1000 - 1000 - 1000 -1000 - 1000 -2000 -2000 -2000 -2000 -2000 -2000 -2000 -2000 -2000 -2000 -2000 -2000 -2000 -2000 -2000 - 2000 -2000 -2000 -2000 -2000 -2000

0 0 0 0 0 0 0 0 0 45 45 45 45 45 45 45 45 45 90 90 90 0 0 0 45 45 45 90 90 90 0 0 0 45 45 45 90 90 90

Temp.

(°F)

E (E6*psi)

0 0 0 - 50 -50 -50 - 100 - 100 -100 0 0 0 - 50 -50 -50 - 100 - 100 - 100 -50 -50 -50 50 - 50 - 50 - 50 -50 - 50 - 50 - 50 - 50 -50 -50 -50 -50 - 50 - 50 - 50 -50 -50

0.2 1.0 5.0 0.2 1.0 5.0 0.2 1.0 5.0 0.2 1.0 5.0 0.2 1.0 5.0 0.2 1.0 5.0 0.2 1.0 5.0 0.2 1.0 5.0 0.2 1.0 5.0 0.2 1.0 5.0 0.2 1.0 5.0 0.2 1.0 5.0 0.2 1.0 5.0

Wm = 0.01 in. PBW PB (psi) (psi)

1074 1345 2478 989 989 2010 904 662 1513 829 1096 2100 743 707 1329 658 331 724 496 456 739 989 988 2010 248 214 249 -495 -507 -817 1983 1953 2467 1493 1458 1520 1102 976 85

1074 1345 2478 989 988 2010 904 885 1607 829 1096 2148 743 894 1644 658 879 1601 638 874 1635 989 988 2010 630 868 1625 615 851 1596 1983 1953 2467 1493 1569 2405 1176 1491 2390

W m = 0.02 in. PBW PB (psi) (psi)

1142 1655 3663 1060 1371 3663 979 1118 3663 904 1376 3139 820 1036 3139 736 717 3139 578 766 2618 1060 1371 3663 333 522 1996 -410 - 181 613 2062 2327 4500 1571 1800 3369 1077 1314 2455

1142 1655 3663 1060 1371 3663 979 1118 3663 904 1405 3148 820 1229 3139 756 1099 3139 749 1188 2618 1060 1371 3663 718 1147 2558 689 1072 2501 2062 2327 4500 1571 1978 3588 1298 1893 3513

W m = 0.03 in. PBW PB (psi) (psi)

1211 1946 4729 1132 1733 4729 1054 1540 4729 970 1630 4090 888 1343 4090 806 1079 4090 649 1042 3591 1132 1733 4729 410 785 3198 - 324 95 1953 2132 2680 6178 1652 2113 5039 1161 1620 3975

1211 1946 4729 1132 1733 4729 1054 1540 4729 970 1677 4090 888 1510 4090 848 1342 4090 841 1471 3725 1132 1733 4729 799 1433 3691 754 1323 3591 2132 2680 6178 1652 2324 5039 1425 2241 4712

P B W = b o r e h o l e pressure initiating fluid invasion into fracture; PB = actual b o r e h o l e b r e a k d o w n pressure inducing a lost circulation. Data: rw = 6 in., u n i f o r m pore pressure thermal e x p a n s i o n coefficient ( ~ ) = 0 . 8 E - 5 ( ° F ) , K T = 0.0116 cm2/sec 5.5 hr after f o r m a t i o n is e x p o s e d to l o w temperature fracture toughness = 1.0 psi*in.**0.5, v = 0.1.

Table 6 shows that the lost circulation pressure for a flawless borehole norally exceeds the in situ stress only by 0.69-3.4 MPa (100-500 psi) if the effective in situ stress (in situ stress minus pore pressure) is 3.4MPa (500psi) since the wellbore conditions are usually hostile as shown in Table 6. Note Wm is 0.254--0.381 mm (0.01-0.015in.) for standard drilling fluids for flawless borehole surface, while W m - W0 must be used instead of Wm for a cracked borehole where Wm is the critical fracture width which allows the mud invasion and W0 is the crack width existing in a formation. The lost circulation pressure increases with the effective in situ stress, but it still ranges only 3.4-6.89 MPa (500-1000 psi) over the in situ stress for a flawless surface with aH = - 6 . 8 9 MPa ( - 1 0 0 0 psi) and av = - 13.8 MPa ( - 2 0 0 0 psi). Lost circulation normally occurs at the borehole with cracks. Figure 15 shows that if the Young's modulus is relatively high, the lost circulation pressure is still 3.4-6.89 MPa (500-100 psi) above the in situ stress even if the borehole surface is not smooth.

CONCLUSIONS

(1) Approximate solutions for fracture width, borehole pressure and fracture length relation are obtained for an inclined wellbore with a crack. The solutions predict a realistic borehole breakdown pressure when a drilling fluid (gelled mud) is used for drilling a well. (2) The new solution predicts that the borehole breakdown pressure depends on formation Young's modulus, borehole size, existing crack size, the width of the closed fracture and the type of drilling fluids, as well as all other parameters involved in the conventional theory. These effects were observed in the field and in the laboratory tests although the conventional theories have ignored them. (3) Thermal cooling, hole inclination and pore pressure build-up affect borehole breakdown pressure, but do not affect it as significantly as conventional theories predict. Thermal cooling lowers borehole

68

MORITA et al.:

BOREHOLE BREAKDOWN--SEMI ANALYTICAL SOLUTION

breakdown pressure for formations with a high Young's modulus. Well angle lowers lost circulation pressure for formations with low Young's modulus. Pore pressure build-up lowers borehole breakdown pressure for formations with low permeability. (4) Predicting borehole breakdown pressure requires accurate information on temperature distribution around a borehole, the Young's modulus, pore pressure, in situ stress, well angle and the existing crack dimension around a well.

E~

tro=--r

-2 [%D

1 --v

az = --

v

I

.JI

rDTdr D-

E

1 --v

aT

E aT 1-v

(A5)

where the plane strain is assumed. Substituting equation (A4) into equation (A5) gives

ao = ~

Ect

[(qr -- 2Tw) - (2Tw + qx)r62 + 2qr In rD] for r o <

e Tw/gr

E~ t Z O = ~ _ V ) [ - - 2 T w + q T ( r ~ 2 - - 1 ) ] r D 2 for r D > e T'/qr. (A6)

Accepted f o r publication 2 February 1995. The tangential pressure at borehole surface is given by

REFERENCES

ao = -

1. Murakami Y. et al. Stress Intensity Factors Handbook, pp. 239-243. Pergamon Press (1987). 2. Yew C. H. and Li T. Fracturing of a Deviated Well. S P E Transactions 285, 429-437 (1988).

E l-v

~tTw.

(A7)

According to continuum theory, borehole breakdown pressure should be reduced by EctTw/(1 - v) if the well temperature declines by Tw. After a few hours, the temperature within the radial distance crossing fracture tip approaches the well temperature Tw for a small fracture. The equivalent surface load in Fig. 4 thus approaches

APPENDIX

New Theory to Predict Lost Circulation Pressure A.I. Temperature effect on borehoie breakdown pressure, fracture reopening pressure and fracture extension pressure Since the thermal expansion coefficient of rock is relatively large, temperature affects borehole breakdown pressure, fracture reopening pressure, and fracture extension pressure. However the effect is localized within several feet around a borehole since rock is a good thermal insulator. Borehole cooling occurs due to fluid circulation. It is significant at the bottom of the well, but it is generally trivial at the casing shoe. Hence, it affects the borehole breakdown pressure if lost circulation occurs around a borehole bottom. The thermal diffusion equation is given by" d2T 1 d T 1 dT (A1) dr 2 r dr K T dt where K r = K/pC: Suppose a wellbore surface cools down due to drilling fluid circulation and is kept at a constant temperature Tw, or

Tl,=,,=T~

f o r 0 < t < oo

TI, =o --- 0

for r w~
(A2)

The solution of equation (AI) with boundary condition (A2) is given by (Conduction o f Heat in Solids, Carslaw and Jaeger, p. 336)

T=T.+~:e-Kr~2Jo(ur)Yo(ur')-Y°(ur)J°(ur')du

u (A3)

Since it is tedious to integrate equation (A2), it is approximated with a simpler form. The temperature distribution with a logarithmic scale show a constant slope for T vs (r/rw) up to a certain radius. The constant slope indicates that temperature distribution can be expressed by a logarithmic approximation up to a certain distance, or

T = T w - q r Inr/rw

for r
T= 0

for r I> r~ e r*/qr

(A4)

where

qT = qT(KTt /r2~)• Several approximate equations have been suggested in the petroleum engineering literatures for the function qT (investigation radius or zone of influence). The continuum mechanics gives the following stress distribution when temperature distribution T(r) is present around a borehole

frD

Ea r - 2 O'r=--'i-~--v D JI rDTdrD

-

E~Tw

[1 + (r/rw) -2] 2(1 -- v)

(A8)

according to equation (A6). The tangential stress around a wellbore induced by uniform horizontal in situ stress is given by

ao = aH[l + (r /rw)-2].

(A9)

Equations (A8) and (A9) show that the stress intensity factor and fracture width induced by temperature are analogous to those induced by in situ stress if a crack is small and temperature sufficiently spreads out. It acts as if the in situ stress is lowered by

EaT, t;~ = -2(I -- v~"

(AI0)

The value ct is relativelyconstant for sandstones and is approx. 8 x E-5 in./in./°F.E normally ranges from I05 to I07 psi for sandstones. Hence ECtTw/[2(l--v)] ranges from 4.7 to 470psi for every 10°F temperature change. Such reduction of borehole breakdown pressure is significantfor tight and low in situ stress formations. However excessive temperature drop may not seriously reduce lost circulationpressure ifa stablefracturereaches beyond the temperature cooling zone. Fracture extension may stop due to its high fracture propagation resistancecaused by the high pseudo-fracture toughness. A.2. Pore pressure buiM-up effect on borehole breakdown pressure fracture reopening pressure and fracture extension pressure Pore pressure build-up normally occurs if formation permeability is similar or smaller than mud cake permeability (1.E-3 to I.E-5 roD). Let's consider the most simple pore pressure build-up problem to check its effect. Assuming one phase flows with constant viscosity, porosity and permeability, the diffusivity equation becomes d2P 1 dP + ar 2 r dr

dp#cdP k r dt

(Al I)

Since borehole pressure fluctuates, we assume an average pore pressure at the interface between drilling fluid cake and rock surface. PI . . . . =P0~

f o r 0 < t < oo

Pl,=0=0

for r,<<.r
(AI2)

The solution of equation (A 11) with the boundary condition (A 12) is given by equation (A3) by replacing T, by P0~. Hence the pore pressure field is identical to the temperature field if nondimensional time krt/qb#cr ~ is used instead of kTt/r ~. The difference, however, is that pore pressure build-up affects wider regions than temperature field since krt/dp#cr~ can be significantly larger than krt/r~ boeause of the relatively small pore fluid compressibility. The solution of equation (AI 1) with the boundary condition (A12) can be approximated by

M O R I T A et al.:

BOREHOLE BREAKDOWN--SEMI ANALYTICAL SOLUTION

P = P0~ - q(tD) ln(r/rw)

for r < r~ exp(Po~/q)

P= 0

for r > rwexp(po~/q)

(A13)

where

q(to) = [2(ln t o + 0.2318) -t 1.154(1n t o + 0.2318)-2]p0~ with t D = k rt/~p~cr 2 for a relatively large tD. The fracture propagation through a formation with low permeability is complex since pore pressure distribution is distributed by filtrate through the fracture surface. Such problems cannot be solved without a complex numerical model. However, some extreme cases can be solved by simplifying the problem, which provides insight into the effect of pore pressure build-up. If fracture speed is very high, the fracture tip pressure, Pr, becomes significantly smaller due to the sudden tip volume expansion. It increases fracture propagation pressure. However, the reduction of lost circulation pressure occurs if the fracture tip pressure Pf becomes sufficiently high because of slow fracture speed or large spurt loss. With sufficient low fracture speed, the fracture tip pressure range is:

differs depending on the area on which borehole distinctly differs depending on the area on which borehole pressure acts, the effects of the two types of pre-existing cracks on wellbore stability are discussed independently in the following section. Narrow pre-existing cracks. Cracks created by thermal or tectonic forces have a width narrower than grain size. They are narrow enough not to allow drilling fluid invasion. However since they do not have any fracture toughness, drilling fluid can penetrate into a fracture with a smaller borehole pressure than for a borehole without any preexisting fracture. Hence, the P w - A / r w - W relation given in Table l should be modified. The crack remains open under in situ stress due to grain mismatching between upper and lower fracture surfaces. The fracture width becomes large if borehole pressure is increased. Let A be the length where the fracture width is affected by borehole pressure change. Then the relations between Pw vs A/r w and W vs A / r w are given by: For A ~rw: Pw= --3trill + O'H2. For 1 . 0 1 r , < A < r , : P~ = [--an, ~ {n/2 + fl x / 1 -- f12 _ arcsin fl }FI

Pore pressure at tip < P f < Pw. Tables 3 and 4 show equations for borehole breakdown pressure and fracture extension pressure with both temperature and pore pressure effects.

A.3. Borehole breakdown pressure from existing cracks Cracks with various sizes and widths exist in formations. When a borehole is drilled through them, some cracks are wide enough to allow drilling fluid to penetrate, while other cracks are narrow enough to prevent drilling fluid entry. Since the stability of a borehole distinctly

RMMS 33/I--F

69

-(~r m - ata)¢flax~ - fl2F2]/t¢[Ix/1 - f12F3]

(A14)

with ( = 2 x ~ / ~ , fl = r~/A 4(1 - v 2) f

W = Wo + ~

~AtymGl + A(o m -- trn2)G2 + 2RwPw~ [G3 + ln(A/r~)]}

(AI5)

where F1 to F3 and G I to G3 are given in Table 1. The above equations show that the zero fracture toughness and pre-existing fracture width significantly reduce the wellbore breakdown pressure.