Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 25, No. 3, pp. 171-182, 1988
Printed in Great Britain
0148-9062/88 $3.00 + 0.00
Pergamon Press pie
Poroelastic Response of a Borehole in a
Non-hydrostatic Stress Field
E. DETOURNAY*
A. H-D. CHENGt
This paper deals with the solution of the stress, displacement, and pore pressure
field induced by the drilling and~or the pressurization of a vertical borehole.
The rock, which is permeated by fluid, is assumed to behave as a poroelastic
material with compressible constituents, following the Biot theory. The
analytical solution is derived in the Laplace transform space, and is transformed to the time domain using an approximate numerical inversion technique. The solution reveals a full range of coupled poroelastic effects which
provide potential mechanisms for delayed borehole instability and shear
fracture initiation inside the rock.
is characterized by a coefficient different from that
of Terzaghi; and (iii) the pore pressure diffusion law
The mechanical response of fluid-saturated porous
is coupled to the rate of change of the volumetric
rocks is characterized by coupled deformation-diffusion
deformation.
effects. These effects, which are caused by the presence
The relevance of poroelasticity to rock mechanics was
of a pore fluid, can be summarized as follows: (i) excess
first discussed by Geertsma [7, 8] thirty years ago, but its
pore pressures are generated by externally applied loads;
significance in various applications has not yet been fully
(ii) the volumetric deformation of rock is controlled by
recognized. The Biot theory is particularly important
an "effective" stress; (iii) the gradient of the pore pressbecause it predicts effects which are not revealed by the
ure acts like a body force which has to be taken into
uncoupled theories. The Mandel-Cryer effect [9, 10] and
account in the equilibrium of the effective stress; and (iv)
the Noordbergum effect during pumping of an elastic
the excess pore pressure is dissipated according to a
aquifer [11] are classic examples in soil mechanics of the
diffusion law. As a direct consequence of these effects,
coupling between the diffusion and deformation prothe volumetric deformation of fluid-saturated rocks
cesses. Demonstration of poroelastic effects in rock
exhibits a sensitivity to the rate of isotropic loading,
mechanics and geophysics have also been made recently.
i.e. the rock appears to be stiffer under "fast" than under
Among other evidences, we would like to cite the
"slow" loading rate. Under rapid loading, the interstitial
dependence of burst pressure on the size of the sample
fluid has little time to escape (undrained condition) and
during the pressurization of a hollow rock cylinder [12];
will resist deformation, thus causing an increase in fluid
stabilization effects in the propagation of shear and
pressure; under slow compression, excess pore pressure
tensile fractures [13-16]; the non-monotonic time varihas ample time to dissipate by diffusive transport (drained ation of the width of a hydraulic fracture [17, 18]; and
condition), and therefore does not contribute to the various effects in earthquakes mechanics such as miapparent stiffness of the rock. These deformation and
gration of aftershocks, induced seismicity, and earthdiffusion processes are fully coupled; any attempt to
quake precursory processes [19].
model the mechanics of the rock-fluid system should not
This paper is concerned with the analysis of the
separate these processes.
various coupled poroelastic processes triggered by the
The simplest consistent theory which accounts for
drilling of a vertical borehole in a saturated formation
these coupled hydraulic-mechanical processes is the Biot
subjected to a non-hydrostatic in situ stress. The anatheory of poroelasticity [1-3]. It differs from the unlytical solution to this problem is derived in the Laplace
coupled theories which have evolved from the earlier
transform space, assuming plane strain conditions in the
work of Terzaghi [4, 5] in the following respects: (i) a plane perpendicular to the borehole axis, and "instanpore pressure generation mechanism characterized by taneous" drilling; the transformed solution is subthe Skempton coefficient [6] is present; (ii) the effective sequently inverted numerically to obtain the solution in
stress governing the deformation of the porous solid time. This solution represents a generalization of the
results of Carter and Booker [20], which were derived
* Dowell SddumbergerTechnolosyCenter, P.O. Box 2710, Tulsa, for the limiting case of incompressible fluid and solid
OK 74101, U.S.A.
t Department of Civil Engineering, University of Delaware, constituents (generally known as the soil mechanics
case). The results due to various loading conditions are
Newark, DE 19716, U.S.A.
INTRODUCTION
171
172
DETOURNAY and CHENG:
POROELASTIC RESPONSE OF A BOREHOLE
examined and analysed, in particular, to provide an
explanation of several phenomena observed around
excavated and/or pressurized boreholes.
Darcy's law
q~ = - xp,.
(5)
and the continuity equation for the fluid phase
POROELASTICITY
The theory of poroelasticity was first introduced by
Blot in 1941 [I]; it has since been re-examined from
various angles [2, 3, 1I, 12, 21-26]. The approach followed here is that of Rice and Cleary [12] who have
provided an elegant reformulation of the Blot theory, in
terms of easily identifiable quantities and material constants. This approach has the further advantage that it
allows a straightforward interpretation of short- and
long-term behaviour of a poroelastic system.
Governing equations
As in the original formulation of Biot [1] (see also Rice
and Cleary [12]), the total stress (7#and the pore pressure
p are chosen as the basic dynamic variables (note that
tension is here taken positive). The corresponding conjugate kinematic quantities are the solid strain e#, derivable from an average solid displacement vector ui, and
the variation of fluid content per unit reference volume,
(. The constitutive equations can be written in terms of
these quantities as follows:
(7# = 2Ge# + ~
P =
2Gv
6#e - o~6#p,
2GB(1 + v.)
2GB2(1 - 2v)(1 + v.) 2
e+
~,
3(1 - 2v.)
9(v~ - v)(l - 2v.)
(1)
(2)
in which 6# is the Kronecker delta, and the basic material
constants are: the shear modulus G, the drained and
undrained Poisson's ratios v and v., and Skempton's
pore pressure coefficient B (ratio of the induced pore
pressure to the variation of confining pressure under
undrained conditions [6]). The parameter a is Biot's
coefficient of effective stress which can be defined as
3(v, - v)
~t = B(I - 2v)(l + v,)"
(3)
Equation (I) reveals that the "Blot effective stress"
(7~= a#+a6#p, is directly proportional to the solid
strain in an identical elasticity relation. Equation (2)
indicates that the pore pressure depends linearly on both
the deformation of the porous solid and the variation of
fluid content.
Besides the constitutive equations (1) and (2), a
complete description of the governing equations for
poroelasticity consists also of
the equilibrium equations
(7#./= O,
(4)
* The requirement of microscopic homogeneity in some micromechanics models (e.g. Nur and Byerlee [30]) is not necessary, provided that two, instead of one, bulk moduli are introduced for the
"non-fluid infiltrated" phase (solid plus pore space) [12, 31].
t A lower bound 3~b/(2 + ~b) for a has been derived by Zimmerman
et al. [34], which is less conservative than the usually quoted lower
bound ~b.
~--~+ q~ l = 0,
ctt
(6)
where q~ is the specific discharge vector, and x the
permeability coefficient, which can be expressed as k//z,
where k is the intrinsic permeability (dimension of length
squared) and/~ the fluid viscosity. In the above, we have
neglected the existence of body forces and fluid sources.
Material constants
A set of five bulk material constants is thus required
to provide a full description of an isotropic rock-fluid
system from a continuum perspective: two elastic
constants G and v [or equivalently the bulk modulus
K = 2G(1 + v)/3(1 - 2 v ) ] , the parameter x which characterizas the diffusive transport property of the system,
and two poroclastic coefficients B and v~ (or the undrained bulk modulus K,) that account for the coupling
between deformation and flow processes. All these constants can be measured experimentally [27-29], but very
few measured values have actually been published.
In a different approach, the material constants can be
obtained by examining the micro-structure of a porous
medium [7, 12, 30, 32, 33]. In fact, the choice of the
above bulk material constants does not postulate any
particular microscopic structure for the porous matrix
besides the constraint that the solid phase is linearly
elastic*. The micromechanical approach, however, deals
with a set of more basic material parameters such as the
bulk modulus of the solid and fluid phase, K, and Ky,
the porosity, ~b, etc. It allows the determination of the
parameters for the continuum description on the basis of
the micromcchanical constants, thus providing insight as
to the admissible range of the bulk parameters. Furthermore the micromechanical approach provides a natural
way for generalizing the theory to semi-linear or nonlinear materials [32, 34].
It has been argued [6, 12, 27] that the realistic range of
variation for B is [0, 1]; for a, [0, !]I"; and for v, [v, 0.5]
(or correspondingly f o r / ~ , [K, oo]). By d ~ i / a g with the
micromechanical parameters, a few limiting cases can be
examined. For instance, the upper bounds for B, v,, and
are simultaneously reached for cases in which both
the fluid and solid constituents arc incomprc~bl¢, i.e.
K~-, oo and g f ~ OO (in practice, these upper bounds
correspond to cases where K, and K / a r e both much
greater than K). This limiting case represents the classical soil mechanics model used for the modelling of clay
consolidation [1]. Note that the Terzaghi effective stress
(7~ = (Tij+p6# is here identical to the Biot ¢ffecIiv¢ stress
(7~. On the other hand, for the consolidation of granular
soil at shallow depth, the fluid is much more compressible than the solid grains, especially when the
entrapped air in the pores is taken into account. We
therefore take K~~ oo, and K/must be interpreted in the
sense of apparent bulk modulus of the gas-fluid mixture.
DETOURNAY and CHENG:
POROELASTIC RESPONSE OF A BOREHOLE
In this case B = Kfl(Ky+ c~K) and K, = Ky/qb + K, and
= 1. This model is identical to that proposed by
Verruijt [11, 35].
In another case, we may consider the fluid to be
infinitely compressible (gas), i.e. Ky= 0. This corresponds to B = 0, /~ = K, while a is unaffected. The
model becomes mathematically equivalent to the theory
of thermal stress (uncoupled thermoelasticity). The
pore pressure is governed by a homogeneous diffusion
equation and interacts with the solid as a body force
through the coefficient a; the solid deformation does not
generate any pore pressure and is uncoupled from the
pressure equation.
Lastly, consider the case of low porosity, tp ~ 0. We
find that K, ~ K .~ K,, ~ ~ (K,/Kp)dp ~ O, where Kp is the
inverse of pore volume compressibility [32]; while B is
unaffected. In this case the pore pressure has no effect on
the solid deformation process. However, since B # 0,
a pore pressure field is nevertheless generated. It is
governed by a diffusion law.
173
where repeated index implies summation. At time t = oo,
the excess pore pressure has dissipated, i.e. p ~ = 0. Then
according to equation (2),
~o= ,re ®.
(12)
It can be seen, by substituting the above expression for
~o~ into equation (7), that equation (7) now reduces to
the classic Navier equation with the drained Poisson's
ratio v.
PROBLEM DESCRIPTION
Let us consider a vertical borehole drilled in a porous
rock formation that is characterized by a nonhydrostatic horizontal in situ stress field, see Fig. 1:
o~ = -(P0-
So),
% = - ( P 0 + So),
Oxy ~ 0,
p =p0.
(13)
Field equations
In the above P0 and So are, respectively, the far-field
The governing equations (1, 2, 4-6) can be combined
mean stress and stress deviator, Po the virgin pore
to yield various field equations for the solution of
pressure. It is assumed that one of the/n situ principal
initial/boundary value problems (see for example the
stresses is parallel to the borehole axis and that x- and
equations proposed by Biot [3], McNamee and Gibson
y-axes correspond to the two other principal directions.
[36], Verruijt [37], Rice and Cleary [12]). For the present
This problem can be analysed by assuming plan e
purpose, they are presented as a Navier equation with a
strain conditions and "instantaneous" drilling of the
coupling term and a diffusion equation, in terms of the
borehole, provided that the time needed to drill over a
displacement vector u~ and variation of fluid content C:
distance equal to about 5 times the radius a of the
G
2GB(1 + v,)
borehole is much smaller than a characteristic time
GV2ui + ~
e,i
3(1 - 2%) ~,t = 0,
(7) t¢ = a2/c. The drilling of the borehole is simulated by
removing at time t --0 the stresses that were acting on
~
cV'~ = 0,
(8) the borehole boundary and setting the pore pressure to
at
7_~ro.
where
To facilitate the physical interpretation of this prob2~:B2G(1 - v)(1 + v,) 2
lem,
the loading is decomposed into three fundamental
c=
,
(9)
9(1 - v,)(v, - v)
modes: (i) a far-field isotropic stress; (ii) a virgin pore
is a generalized consolidation coefficient [12]. Alterna- pressure; and (iii) a far-field stress deviator. Not,
tively, equation (8) can be written in terms of pressure that modes 1 and 2 above are axisymmetric, while mode
3 is asymmetric. Denoting by the superscript (i), the
Op 2gGB2(1 - 2v)(1 + v,) 2
stress induced by the loading mode i, the boundary
at
9W.:
v2P
2GB(1 + v,) t~e
=
3(1 - 2v,)
~-t"
(10)
For boundary-value problems characterized by the
application of constant boundary conditions, the initial
(t = 0 + ) and final (t = or) solution are simply obtained
by solving an elasticity problem with undrained and
drained elastic constants, respectively. Consider first the
conditions at t - - 0 +. Initially upon application of
the boundary conditions, fluid has not yet escaped from
the pores, i.e. ~ = 0, and equation (7) reduces to the
classic Navier equation with undrained Poisson's ratio
v,. Combination of equations (1) and (2) indicates that
the initial excess pore pressure p0+ is given by
B 0+
p0+ = _ -3 o k , ,
(11)
l
- t r . ffi P . + So
Po
- o u = Po - So
T
Fig. 1. Problem definition.
174
DETOURNAY and CHENG:
POROELASTIC RESPONSE OF A BOREHOLE
conditions at the borehole wall for each of the loading
modes can be written as follows.
•
o-~ = 2GA ~
2~1 -fi
+
rp dr - 2tlp,
o-,0 = 0.
mode 1:
o-o)
= P0
rr
o-~ = 0
p(I)
= O;
(14)
• mode 2:
.(2) =
rr
The above results are formally identical to those obtained by Carter and Booker [20] for the case of incompressible constituents, except that the material
coefficients are now generalized. To get Carter and
Booker's solution, we simply use B = 1.0 and vu = 0.5,
which results in the reduced material constant definitions
0
C=
.~)=0
(15)
p(2) = - - P 0 ;
• mode 3:
O.(3)
,,
(20)
= - So cos 20
o-~) = So sin 20
pO) = 0;
(16)
where r and 0 are the polar co-ordinates, defined in
Fig. 1. Both the induced stress and pore pr~sure vanish
at infinity. Solutions for the induced stress, pore pressure, and displacement axe derived next for each of the
fundamental loading modes.
2rG(1 - v)
1 -2v
1 -- 2v
'
~/= 2(1 -- v)"
(21)
The solution of the first two loading modes can now
be obtained by introducing the appropriate boundary
conditions.
Loading mode 1. For this loading mode, equation (17)
yields the trivial solution p = 0. In this case, we obtain
the classical Lain6 solution in elasticity
2Gu~ ')
a
Pod
r '
6(0
a
2
rr
w
~
n
_
Po = r-~'
o-~
a2
(22)
Po = - - 7 "
ANALYTICAL SOLUTION
Axisymmetric loading
Calculation of the displacement, stress, and pore
pressure induced by the fundamental loading modes 1
and 2 can be carried out in a parallel manner. Indeed,
it can be demonstrated [12] that, in the case o f a borehole
in an infinite domain and under axisymn~tric loading,
equation (10) can be uncoupled to give a homogeneous
diffusion equation
O2p l a p
lop
0r 2 + r Or = -c- -0t
.
(17)
The pore pressure can therefore be solved independently
of the other quantities. After obtaining the general
solution of p, the displacement can be calculated by
solving equation (7), to give
rp dr,
(18)
Po
~(~)
Ko(fl)'
(23)
3(v, - v)
1 - 2v
= a ~
r/= 2B(I - v)(l + v.)
2(I - v)'
(19)
is a portxlastic coefficient(the physical range of variation oft/is [0--0.5]).It should be noted that in the above
the displacement has been required to vanish at infinity.
From equation (18) the polar stress components can
be derived easily as follows
lff
rp dr,
of order zero,
= r
and a
follows
from equations (18) and (20) that the Laplace tmmforms
of the displacement, stress and flux fields are
where r/, given by the expression
I
sP (2) _
where Ko is the modified Bcssel function of second kind
u, = A -r1 +-~~lf~r-
o-,, = - 2GA -~ - 2~ -~
This result could have been also directly d ~ v e d from
the consideration that the rock deformation in this case
is entirely associated with the deviatoric strain. As a
consequence, there is no mechanism for pore pressure
generation, and the displacement and stress field are
independent of the bulk modulus of the rock and also of
time.
Loading mode 2. In this case, the constant A in
equations (18) and (20) vanishes since the total radial
stress o-,, at the borehole wall (r = a) is equal to zero.
The pore pressure field is solved by taking the Laplace
transform of equation (17). Solving this ordinary
differential equation with the boundary condition (t5),
we obtain
I ~K,(~)
2G'~[2------~)= 2q L
'~')
Po a
[a K,g)
--=
Po
-2~/[.r p - - ~ )
po
~Ko(/~)
ar ~Ko(~)
KtO) ] '
(24)
a K,t ) ]
7 p"-~)J'
a ~ It,O) ± K,(O]
s a ~ 2)
pox
flK,(~)
Ko(fl ) "
/]--~-,
~_],
(25)
(26)
(27)
DETOURNAY and CHENG: POROELASTICRESPONSE OF A BOREHOLE
Deviatoric loading
For loading mode 3, the solution is also obtained in
the Laplace transform domain. Equations (7) and (8)
can be Laplace transformed and written in polar coordinates as
l - v , 0~
1 - 2v. Or
10o3
B(l+v~)0~"
~3(1 - 2v~) Or
r O0
l - v . 1 c3~ 0o3
1 - 2 v , r 00
Or
0,
(28)
B ( l + v . ) 1 0~"
= 0,
3(1 - 2 % ) r 00
(29)
02~" 1c9~" 1 02~" s~.=O '
(30)
~ r 2 + r ~ r -] r2 ¢~0 2
1 - 2vu
a
a3
2(1 - v.----~C2 -r + C3 r-~.
where 03 denotes the rotation of the displacement field.
Using symmetry considerations, it can be argued that
the dependence of the displacement and stress upon the
polar angle 0 is of the following form
sill B2(I -- v)(l "I-v.)2
C,K2(~)
So 9(1 - v.)(v. - v)
B(1 + v.)
a2
-t ~ - ~ C2 ~ ,
s•,. B(l+v.)
1
1 - - Vu
1 - v . d J ~ _ 2 li>
1 - 2% dr
r
l-v, g
1-2% r
r2d2~
B(I+%) d2
=0,
3(1 -- 2v.) dr
ld#
2 dr
~ r 2+
B(l+v,)
rd~_(S_r2
dr
\c
+ 4 2~=0.
(34)
)
B(1
1 - 2v. ¢., a~
1"- ~ --.2 r 2.
~oa =
B(I+%) I
2
]
3-'~-1~ ~'.) C, Ka(~) + ] K2(~)
a
a 3
+ C2 - + C3 -r -3 '
r
2GsOo
Soa =
R.M.M.S. 2513--F
2B(I + v.)
a
3fl2(I-- v.) C,-r K2(~)
r 4'
1+
K2(~)
]
(41)
(42)
So -
3
]
3(1 - v.) C, ~ Kt(~) + ~ K2(~)
1
a 2
a4
2(I - v.-----C2
---)~ -3C3 ~.
(43)
C, = - 12~0(1 - v.)(v. - v)
B(1 + vu)(D2 -- D,)'
C 2~
C 3
4(1 - vu)D2
D 2 -- DI
~(D~ + D,) + 8@, - v)K2(#)
fl(D2 -- DI)
'
(44)
where
D, = 2(v.- v)K,(#),
D2 = ~O(l - v)K2(10).
(45)
(37)
In equations (35)-(37) we have taken into account
that 2, # and g must vanish at infinity. It is of
interest to point out that equation (37) contains a term
that does not appear in the solution with incompressible
constituents [20].
The displacement components are found by solving
equations (36) and (37)
2GsO,
a 4
a2
C2 ~ - 3C3 - -
s~,e 2B(l + v.) [1
(35)
After some manipulation, equations (32) and (33) can
also be solved to yield
a2
s # ffi C~ ~ ,
(36)
+ v.) C~K2(~)
3 ( 1 - v,)
]
a 4
Equation (34) can be easily solved for
s~ = C~K2(~).
K2(~)
(32) The three constants, Ca, C2 and Cs, are obtained from
the boundary conditions (16). They are
(33)
r
~2
-4- 3C 3 ~'i,
(31)
=0,
3(1-2v.)
~K,(~)+
3(1 - v.) C, ~ Ka(~) +
=
in which 2, /~, # , 0,, 0o, ~,,, ~ , ~ , and P
are functions of r and s only. Substituting the above
expressions into equations (28)-(30) produces the set of
ordinary differential equations
(40)
[1
,<,+v.)[l
So -
(~'% ~,',, af~>,~,, ~gDpo)) (2, g, 0,, ~,,, ~ , P)cos20,
(o3°), a~3), O~)) = ( # , 0o, ~,o) sin 20,
(39)
In the above we have redefined the constants C~ and C~
into C, and C2. The stress and pore pressure can now be
deduced easily from equations (36)-(39):
~-o = 3 ( I - v , , ) C '
C
175
(38)
RESULTS IN TIME DOMAIN
The numerical results in the time domain can be
obtained by applying an approximate Laplace inversion
to the analytical expressions derived in the preceding
section. There exist a great variety of numerical Laplace
inversion methods (see the bibfiography compiled by
Piessens [38]). The numerical results below are carried
out using the method of Stehfest [39], which has received
high marks [40] for its accuracy, efficiency and stability.
The method is based on sampling inversion data according to a delta series. The approximate solution in time
is given by the formula
In2 ~
-/ In2\
c.f{,n-r
),
(46)
176
DETOURNAY and CHENG:
POROELASTIC RESPONSE OF A BOREHOLE
with the coefficient C, given by
1.00
C, = ( - - 1)"+N/2
A 0.80
k NI2(2k )!
min(n,~N/2)
k = ( ~ ~)/2( N / 2 -- k ) [ k ! ( k - 1)!(n - k)!(2k - n)["
(47)
" 0.60
The number of terms N in the s e r i e s is even and is
typically in the range of 10-20.
We now selectively examine some of the numerical
results for mode 2 and mode 3 loading. Note that the
background values for the stress and pore pressure have
now been added to the induced components (e.g.
P = P0 + p(2)for mode 2), and as a result the superscript
(i) has been dropped.
0.40
r/a
15
20
0.20
o.oo! i
10 -2
f i111111
10"1
i
i tlf)lt[
i
i iiii111
1
i I illlll 1
10
102
I
i ,ll,lq
I )111111
10 a
t0 (
T I M E (t*)
Fig. 3. Pore pressure history at various r/a for mode 2.
Mode 2
Figure 2 shows isochrones of the normalized tangential stress o ~ / q p o in the vicinity of the borehole (the
stress a~/~lpo is independent of any material properties,
and is a function of the two normalized variables p = r / a
and t* = c t / a 2 only). It can be seen that, except at the
borehole wall, the tangential stress history at each point
is first compressive (negative), then tensile (positive). The
large time asymptotic value a ~ , given by the expression
stress, ¢rss+ p = ( 2 r / - - l)p 0 (0~<~/ ~<0.5). I t is a l s o o b s e r v e d , on the basis of equation (24), that there is no
movement of the borehole wall induced by mode 2
loading, although there is a non-zero, inward radial
displacement inside the medium. The large time asymptotic value is given by
2---6-
, ~ = qp0 (1 + ~22),
(48)
is reached rapidly in the vicinity of the borehole and
instantaneously at the borehole wall. (The asymptotic
expression (48) can be obtained by setting p = -P0 in
equation (20) or inverting equation (26) in the limit of
s --, 0.) At large distances however, the transition is slow
and the asymptotic value is physically impossible to
reach, because it corresponds to the total drainage of
pore fluid from the domain and the vanishing of the pore
pressure everywhere. The small time asymptotic solution
can be obtained by an analytical inversion of the expansion of the Laplace transform solution. This result,
together with results for other quantities, are presented
in the Appendix.
From Fig. 2 we also note that the stress concentration
at the borehole wall is tensile in terms of total stress,
a~ = 2~/p0, but compressive in terms of Terzaghi effective
-
"
This value is reached rapidly in the vicinity of the
borehole.
The pore pressure and radial flux histories are plotted
in Figs 3 and 4 for various distances from the borehole.
The initial flat parts of the pressure curves susgest that
the drainage area propagate with a relatively steep front.
The flux movement is inward and is maximum at the
arrival of the front. T h e maximum rise of the flux
diminishes rapidly with distance. At the borehole, the
flux is initially infinite and then experiences a continuous
decay. For early times, according to the asymptotic
solution in the Appendix, the flux varies inversely as the
square root of time.
Mode 3
Isochrones of the induced pore pressure variation with
distance from the borehole are plotted in Fig. 5, for the
parameters v = 0.2, v. = 0.4, B = 0.8, and the direction
0 = 0, n. Upon the instantaneous drilling of the bore-
2.0
0.t
1.00
¢J9
~c
I-
c~
A
1.0'
= 1.5
0.40
..J
IJ.
W
0.20
i
i
r
2
R A D I U S (r/a)
Fig. 2. Isochrones of the tangential stress variation with radius for
mode 2.
10 -2
10 -1
1
10
T I M E (t °)
10 =
10 s
Fig. 4. Radial flux history at various r/a for mode 2.
10 (
DETOURNAY and CHENG: POROELAST1CRESPONSE OF A BOREHOLE
1.so |,o+
......
177
2.40
LU
2.00-
I.U
o
1.6o.-I
<
o.®V
1.40
1.20
1. o
i
i ,l.llq
10 .4
i
10 "s
J JJlln[
R A D I U S (r/a)
7r for mode 3 (v = 0.2, v= = 0.4, B = 0.8).
4
a2
/,o+ = 3 SoB(l + v.) ~ cos 20,
1+3
(50)
S0cos20,
(51)
which is sketched as a dashed line in the figure. The
stress concentration at the borehole wall, r = a, however
•
r
It,,ll,
1
l
10
l
I IIIII11111'1~]~1
10a
10 3
Fig. 7. Radial displacementhistory at r/a = 1, 0 = 0, ~ for mode 3
1 -
lira a0e(a, 0; t) = - 4 ~
v=
So cos 20.
(52)
¢...0 ÷
As a result, at very small time (t* < 10-2), the peak of
the tangential stress is actually located inside the rock
and not at the wall of the borehole, as predicted by the
elastic analysis. At greater times, the tangential stress
decreases monotonically with d/stance, and the stress
concentration at the wall increases towards the longterm elastic value 4Socos20. It is also interesting to
observe that significant deviation of a~ from the elastic
distribution takes place only in a small region near the
borehole. It is thus appropriate to describe the poroelastic mechanism as a skin effect. It should be noted
that, in contrast to ~00, the radial stress component o,,
experiences little variation.
Finally, the history of the radial displacement at the
boundary point 0 - 0 ° is plotted in Fig. 7. The radial
displacement varies from (3 - 4v=)aSo/2G at small times to
(3-4v)aSo/2G at large times. Since the radial displacement at the boundary is proportional to cos 20, it can
be seen that the existence of a far-field stress deviator
produces a time-dependent opening in the direction of
the minimum compressive in situ stress ("reverse consolidation") and a progressive closure in the perpendicular direction. This phenomenon was first noted by
Carter and Booker [20].
APPLICATIONS
-2.00-
As an application of the solution derived above, we
examine the role of poroelastic processes on the condition leading to formation breakdown during pressurization of a borehole, and on the stability of a borehole
following drilling.
09
¢n
,~
l,J.llff
is instantaneously reduced to
is generated, which is plotted as a dashed line in the
figure. The enforcement of the boundary condition p --- 0
is, however, responsible for a steep radial gradient of the
pore pressure at early times, which is associated with a
rapid drainage of fluid at the wall. (Note that since the
pore pressure varies with the polar angle 0, a tangential
flow also develops.)
Rapid drainage of the rock near the borehole has a
direct impact on the stress concentration. Indeed, at the
borehole, the rock is characterized by the drained elastic
modulus, while the rock further away is characterized by
the stiffer undrained modulus. As a result of this stiffness
contrast, the borehole is partially shielded, at early times,
from the stress concentration. This effect is well illustrated in Fig. 6 where the isochrones of the tangential
stress variation with radial distance have been plotted
for the direction 0 -- 0, ~. At the instance of drilling, the
tangential stress at r > a is given by (see Appendix)
= -
l
10 "1
(v = 0.2, v= = 0.4, B = 0.$).
hole, an undrained pore pressure distribution
0.
t,l,lt,J
T I M E (t*)
Fig. 5. Isochronesof the pore pressure variation with radius at O~-0,
aoo
r
10 "a
-3.00-
Borehole pressurization
Z
-4.00
l oo
1.'1o
1. o
1. o
1.so
RADIUS (r/a)
Fig. 6. Isochrones of the tangential stress variation with radius at
0 -- 0, g for mode 3 (v ~ 0.2, v, = 0.4, B -- 0.8).
This problem has its root in the hydraulic fracturing
technology used by the petroleum industry for stimulating wells. In this technique a section of a borehole is
sealed-off, and fluid gradually injected to raise the
pressure in that interval. At "breakdown", the fluid
178
D E T O U R N A Y and C H E N G :
P O R O E L A S T I C RESPONSE O F A B O R E H O L E
pressure in the borehole is large enough to initiate a
fracture in the hydrocarbon formation.
A rigorous study of the breakdown process should be
carried out from the perspective of fracture mechanics,
by analysing the requirements for the propagation of a
pre-existing flaw at the borehole [41-45]. However, it is
often simply postulated that breakdown takes place
when the Terzaghi effective tensile stress at the borehole
wall is equal to the tensile strength, T, of the rock, i.e.
a00 + P = T.
(53)
The stress concentration induced by pressurization of
the borehole (p =Pw and tr,, = -pw at the wall) in the
presence of in situ stresses can be deduced by superposing the three modes of solutions analysed earlier.
Neglecting momentarily the existence of an in situ stress
deviator So, it can be deduced, from equations (22) and
(26), that the tangential stress at the borehole wall is
a~=-2Po+2tlpo+(1-2~l)p~;
r=a.
(54)
It should be noted that the above equation does not
impose any restrictions on how the borehole pressure
varies with time. In other words, there is no time lag
between the borehole pressure and the resulting stress
concentration (this will not be true, however, in the
presence of a filter cake). If a far-field stress deviator So
exists, the stress concentration due to this loading has to
be added to equation (54). This value varies from the
expression in equation (52) at small time to the elastic
value 4S0 cos 20 at large time.
Taking into account equations (53) and (54), and
assuming that the transient deviatoric stress effect has
been damped out such that the long-term elastic value
prevails, the breakdown pressure is given simply by the
expression
Pu =
T + 2P0 - 4S0 - 2r/p0
2(1 - r/)
(55)
This expression, first derived by Haimson and Fairhurst
[46], represents a lower bound for the breakdown
pressure.
Another case of interest is when the fracturing
fluid cannot penetrate the rock (sleeve fracturing for
example). Through appropriate combination, the predicted breakdown pressure Pb,, is
Pb, = T + 2/'o - 4S0 - 2~/p0- (1 - 2r/)p,,
(Terzaghi) stress concentration at the borehole wall
becomes more compressive with reduction of well pressure; with the potential of causing compressive failure at
the wall. (It is generally accepted that compressive
failure, as welt as tensile failure, is controlled by the
Terzaghi effective stress [49, 50].) These concepts have
been further extended by assuming the rock to behave as
a poroelasto-plastic material [51-53].
We are here interested mainly in investigating the role
of the poroelastic effects associated with mode 3 loading,
in the mechanism of borehole failure following drilling.
It will be demonstrated that poroelastic mechanisms can
be responsible for a delayed borehole collapse. In addition, the ensuing analysis also suggests that failure can
be initiated at a small distance inside the rock, rather
than at the borehole wall.
Consider first the boreho!e stress concentration caused
by mode 3 loading. As noted earlier, the short-term
solution for the tangential stress at the borehole wall,
following drilling, is given by the expression (52). The
stress concentration will then increase monotonically
with time to the value predicted by the theory of
elasticity, see Fig. 8. The magnitude of the variation of
the maximum stress concentration is of the order of the
in situ stress deviator So. The characteristic time for this
variation is about tc ~ a2/c (t* ~ 1), but it must be noted
that the presence of a filter cake can increase t~ as much
as a factor of 10. (An upper bound for this characteristic
time can be obtained by solving mode 3 loading with a
no-flux boundary condition at the borehole. This solution was obtained by Carter and Booker [20] for the
case of incompressible fluid and solid constituents.)
It is worth reiterating that, for constant boundary
conditions, mode 3 is the only loading component to
introduce time-dependent variation of the stress concentration. Indeed, the borehole response to mode 1 is
purely elastic, while the maximum stress concentration
induced by mode 2 is reached instantaneously. This
time-dependent increase of the stress concentration suggests a possible mechanism for delayed instability of
boreholes. In other words, if the compressive strength of
the rock is between the small time and large time
asymptotic effective stress concentration values, the collapse of the borehole can be delayed for a time as much
as lOa2/c. It is possible, however, to undertake a more
(56)
where Pl is the pore pressure at the borehole wall. Taking
a specific example e.g. Po/T = 14, So/T = 2, po/T = 8,
~/=0.25. We find that p u / T = 11.4 and p ~ , / T = 13.0
(assuming that p,. =P0). The expression (56) for nonpenetrating fluid is seen to yield higher values for the
breakdown pressure than equation (55).
Failure initiation around borehole
Previous poroelastic considerations of borehole failure have generally focused on the change of effective
stress that accompanies production of the well [47, 48].
Application of the mode 1 and mode 2 solutions
discussed above indicates in fact that the effective
Z
hI.-Z
LO
3.5-
8
oJ
er
3
10-3
1'0"
1'0-'
1'00
1'0'
102
T I M E (t')
Fig. 8. Stress concentration history at 8 - - 0 , n for m o d e 3 (v --0.2,
v, -- 0.4, B = 0.8).
D E T O U R N A Y and CHENG:
POROELASTIC RESPONSE OF A BOREHOLE
the isochrones of S vs P" in Fig. 10 reveal a great deal
of variation with time. This is a direct consequence of
significant pore pressure change over a considerable
region induced by mode 3 loading. It is of interest to
note that the elastic stress profile given by
2.00
_
1 -1
S
0.50
I
0.00
0.50
1.00
1.50
MEAN PRESSURE (P/So)
2.00
Fig. 9. Isochrones o f stress profiles S - P in the radial direction 0 = 0,
n for mode 3 (v ffi 0.2, v, = 0.4, B ffi 0.8).
exhaustive analysis of the conditions leading to failure
initiation, by considering also the evolution of the state
of stress in the vicinity of the borehole in conjunction
with a given failure criterion. A condensed way of
presenting the stress evolution associated with the failure
criterion is to plot, along a given radial direction and at
selected times, the maximum planar shear stress S,
defined by
S
= ~1 x/(a00 - a.)2+ 4a~0,
(57)
as a function of the mean planar compressive stress P
1
P = - ~ (a~ + a,,).
(58)
Figure 9 shows the parametric relationship between S
and P along 0 = 0 for various times t*. Figure 10
presents the same plot with the Terzaghi effective mean
(compressive) stress P " = P - p substituted for P. The
stress profiles in these figures encompass the range from
r ffi oo (corresponding to the left ends of the curves
where P ffi 0 and S---So) to the borehole wall (represented by the right ends of the curves which terminate
at the intersections with the line P = S).
Figure 9 reveals little difference between the curves,
except in the region close to the borehole. By contrast,
2.00
A
~t*
= 10-s
,.1.o4
rr 1.500
lO8
-
...
1.14
~';~
1.20 -
-
"
-
- 1,"~
................
.o
.
.~. . . . . . . . . . . . . .~. .
,
ic
"
~ 1.1~-
0.50
o.oo
o. o
i. o
179
1. o
zoo
TERZAGHI EFFECTIVE MEAN PRESSURE (P"/S,)
Fig. 10. Ig~chrones o f streu profiles S - P " in the radial direction
0 --0, g and s t r ~ s history at ~leeted points for mode 3 (v = 0.2,
v. = 0.4, B = 0.8).
P
3P 2
(59)
where P should be interpreted as P", serves as the lower
envelope for the family of curves. Another view of the
variation is given by displaying the stress history at
selected points. These stress histories are shown as
dashed lines in the same Fig. 10; they represent the
deviatoric vs mean compressive stress relations at
selected r for 0 ~< t < oo.
It is not our intention here to perform an exhaustive
study of the conditions of impending failure, but rather
to test a conjecture that shear failure can actually initiate
inside the rock, rather than at the borehole wall [54]. The
following assumptions are made: (i) the failure criterion
corresponds to a Mohr--Coulomb envelope, (ii) failure is
controlled by the Terzaghi effective stress, and (iii)
failure is independent of the out-of-plane normal stress
a,,. Within the constraints of these assumptions, the
failure criterion can be expressed as simply as
K,-I(
S=Kp+I
'~,
q"
P" + K p - 1]
(60)
where q= is the uniaxial compressive strength, arid K~ the
passive coefficient, given in terms of the friction angle ~b
by
1 + sin
Kp = 1 - sin ~b"
(61)
Considerations of failure require superposition of
all the stress modes. Consider the following case characterized by the absence of mode 2: Po= 7, So = 1,
p o = p , = 4 . The corresponding stress profile ( S - P " )
along the x-axis is shown in Fig. 11 for different values
of the normalized time t*. Figure 11 differs from Fig. 10
in that the stresses for mode 1 have now been taken into
account.
The straight line in Fig. 11 represents a MohrCoulomb criterion criterion characterized by a friction
angle 4~ --- 30° and a uniaxial strength q, = 10. A profile
above the line implies failure. As indicated on that figure,
these parameters have been chosen in such a way that a
state of impending failure exists at the borehole wall (at
0 = 0) according to the elastic solution (i.e. the MohrCoulomb line intersects with the right end of the elastic
stress profile). The figure clearly indicates that failure
can indeed initiate at some distance away from the
borehole, i.e. not at the borehole wall as is predicted on
the basis of the elastic stress distribution. According to
the stress history curves (dashed lines), failure occurs at
a distance ranging from 5 to 10% of the borehole radius
away from the wall. The poroelastic effects associated
with mode 3 thus provide a potential mechanism for the
formation of borehole breakouts, a pervasive feature of
deep wells [55-57].
180
DETOURNAY and CHENG:
6.00
A
° 5.00-
Mohr-Coulomb(0 = 30°)
t'= 10-' ., _5
~ "
~ 3.00- ~
u~
2.00-
POROELASTIC RESPONSE OF A BOREHOLE
p=1.00
J
~. . . . .~---.
~
"
Elastic
1.00-3.00
3. 0
4.60
4. 0
s.oo
TERZAGHIEFFECTIVEMEANPRESSURE(P"ISo)
Fig. I 1. Determination of the location of shear failure initiation
(v ffi 0.2, v, = 0.4, B = 0.8).
The impact of varying the failure parameters q= and
could be analysed graphically in the diagram of Fig.
11 (note that a change in the compressive strength qu
causes the Mohr-Coulomb envelope to shift up or down,
parallel to itself, while variation in the friction angle is
reflected by a change in slope of the limiting line). For
example, if the friction angle is zero, failure can initiate
away from the borehole or at the wall depending on the
compressive strength of the material. Note finally that
increase in friction angle tends generally to promote
initiation of failure inside the rock medium.
could be initiated inside the rock, along a radial direction
perpendicular to the in situ major compressive stress,
rather than at the borehole wall, as is predicted on the
basis of an elastic analysis. This finding gives some
support to the conjecture that some borehole breakouts
actually initiate inside the rock, rather than at the
borehole wall.
As a final note, we observe that the poroelastic
solution derived in this paper also applies to moderately
deviated wells (i.e. for cases where the vertical gradient
of the in situ stress and pore pressure can be ignored).
An anti-plane strain component must however be added
to the solution, as the drilling of a borehole inclined to
the principal stress directions causes the removal of
longitudinal shear stresses along the hole surface. This
anti-plane solution causes a local warping of the plane
of solution, associated with a shear strain Erz. However,
the anti-plane component has no influence either on the
volumetric strain and or the mean stress; this component
is therefore time-independent and purely elastic.
Acknowledgements--The authors would like to acknowledge with
thanks the management of Dowell Schlumberger for allowing publication of this material. They also want to thank Dr J-C. Roegiers for
his continuing support of this project.
REFERENCES
CONCLUSIONS
In this paper we have presented an analytical solution
to the problem of a vertical borehole drilled in a
fluid-saturated poroelastic formation subjected to a nonhydrostatic far-field stress. The solution presented in the
paper is a generalization to the poroelastic model of Biot
of a solution derived originally by Carter and Booker
[20] for the limiting case of incompressible fluid and solid
constituents (soil mechanics case).
The poroelastic solution was obtained in closed form
in Laplace transform space, and numerically inverted,
using the Stehfest algorithm, to retrieve the dependence
of the solution upon time. However, asymptotic solutions for small time are derived explicitly as a function
of time (see Appendix).
The paper has focused essentially on the poroelastic
effects associated with the existence of a far-held stress
deviator So (mode 3 loading). We have proven that the
undrained elastic solution for the tangential stress and
the undrained pore pressure distribution is not the
proper short-term solution in the neighbourhood of
the borehole. In particular, it was found that the shortterm stress concentration associated with mode 3 is
4(1 - v,)[(l - v)So cos 20 and not 4S0 cos 20 as predicted
by an elastic analysis.
We have also investigated the conditions of failure
initiation based on the poroelastic stress distribution, by
assuming the rock to be characterized by a cohesivefrictional strength and failure to be controlled by the
Terzaghi effective stress. It was found that shear failure
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APPENDIX
Small time asymptotic solutions
On the basis of the expansion of Laplace transform solution of modes
2 and 3, it is possible to derive the small time asymptotic solution for
the region near the borehole.
Mode2
P 1-. ae~c
-~
-
Po
(r -- a)2~
x exp(
,
(r
-~JL4~
1 eric -r-- - a
Deformation and Failure of Granular Materials, Proc. IUTAM
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pp. 167-170. Balkema, Rotterdam (1982).
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o~ = 27
P0
r
exp
a2
1 a
-(:-)
1 eric r -- a
o,, = _ 2r/
o0
4ct
r
apo =
.
exp -
I effc ~
:o.,
+ eric
~-7
~
b/TL~/~-~
- a
exp(
-~
a2
4ct
~
.
(r --a)l~
~
)
/
.
182
DETOURNAY and CHENG: POROELASTIC RESPONSE OF A BOREHOLE
Po
raq' = _ f ~ {a ~ c t
exp(
7al-~
a,o= I - - 1 - 2 n2 + 3 ¢~41 sin 20.
(r a)2'
- ~__~
) + 41erfc(r-a~\~/~t./
So
/ (r-a)~'~
32. Lx/~oxp[
~-./
2Gu,
(A5)
This solution indicates that the maximum (tensile) stress concentration 2ripeis reached instantaneously at the borehole. Also, the radial
flux at the boundary varies as the inverse of ~ for small time.
2Gue
aS°
SoN
Mode3
~/+
[-
aS° = L4(l - v~) r
- ( r - 1) erf¢(~4c~)]t'
= ~
~
~
co.,28.
(A6)
[
-~
cos 20.
(AI0)
°'1
(AI 1)
- 2(1 - 2v,) a _ ~
3
x
k r
exp (
(A9)
sin 20.
~/ r ~/ ~ct
)jco.2o.
(r - a)v~-I
(AI2)
a4
- 4~ +3
(AT)
~o=[-,+4--~--Z~_vN/~erfc(r-a]-\
,_~,/ ~]cos20. (A8)
Vu -- IJ
a
3
a4
The previous equations show that the undrained elastic solution is
the correct short-term solution for the d/s#ncem~t liekl and the stras
components ¢,, and o,e. The undrained elastic soluti~ is however not
the proper short-term solution, in the near bore-bole ~ n l for the
tangential stress ow and for the pore pretmure p.