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Poroelastic solutions in transversely isotropic media for wellbore and cylinder
\
Int[ J[ Solids Structures Vol[ 24\ Nos 23Ð24\ pp[ 3894Ð3818\ 0887 Þ 0887 Elsevier Science Ltd[ All rights reserved Printed in Great Britain 9919Ð6572:87:,*see front matter
Pergamon
PII ] S9919Ð6572"87#99090Ð1
POROELASTIC SOLUTIONS IN TRANSVERSELY ISOTROPIC MEDIA FOR WELLBORE AND CYLINDER Y[ ABOUSLEIMAN Department of Civil Engineering\ School of Engineering and Architecture\ The Lebanese American University\ Byblos\ Lebanon
and L[ CUI Rock Mechanics Institute\ The University of Oklahoma\ Norman\ Oklahoma 62908\ U[S[A[ "Received 0 September 0886 ^ in revised form 01 February 0887# Abstract*This paper presents closed!form solutions for the pore pressures and stress _elds for the inclined borehole and the cylinder\ induced by boundary stress perturbation in an anisotropic poroelastic medium[ The governing equations for the transversely isotropic poroelastic materials under the cylindrical coordinate system are presented[ The solution for the inclined borehole\ subjected to a three!dimensional far!_eld anisotropic stress _eld\ is derived with the borehole generator coinciding with the material axis of symmetry ^ while the solutions for the cylinder are obtained under various loading conditions that are encountered in laboratory testing procedures[ The analysis shows\ in addition to the e}ect of time on stress and pore pressure variations\ that poromechanical anisotropic material coe.cients also play an important role in calculating the in! plane stress _elds\ which is not the case using the classical anisotropic theory of elasticity[ Þ 0887 Elsevier Science Ltd[ All rights reserved[
0[ INTRODUCTION
The theory of anisotropic poroelasticity was formulated by Biot "0844#\ in an attempt to generalize consolidation theory[ Later\ Biot and Willis "0846# extended the theory to describe various poromechanical properties of an anisotropic ~uid saturated medium[ In their work\ the outlined experimental procedures for the determination of the directional poromechanical properties are extremely di.cult to achieve\ if not impossible[ Following the interpretation of Rice and Cleary "0865# of the poroelastic material constants\ Thomp! son and Willis "0880# reformatted the anisotropic poroelastic constitutive relations\ in which the parameters that appear are given explicitly in terms of drained "elastic# and undrained compliances\ and the generalized tensors of Skempton|s pore pressure coe.cient\ Bij\ and Biot|s e}ective stress coe.cient\ aij[ Recently\ these parameters have been cast in a practical model for laboratory measurement "Cui\ 0884 ^ Cheng\ 0886#[ With these latest developments in material characterizations\ the application of the anisotropic poroelastic theory to engineering problems becomes more amenable[ Although geomaterials exhibit many forms of anisotropy\ closed!form solutions for anisotropic por! oelasticity are almost non!existent in the literature[ Recently\ the transverse anisotropic poroelastic solutions for the one!dimensional consolidation "Cui et al[\ 0885a# and the two! dimensional Mandel|s problem "Abousleiman et al[\ 0885a# were derived[ These solutions may not _nd a wide range of engineering applications\ but they can serve as benchmarks validating numerical algorithms such as the _nite element method "Cui et al[\ 0885b#[ In this work we extend the transverse isotropic poroelastic solutions to the cases of the inclined borehole and cylinder geometries[ Stability of wellbores in deep drilling remains one of the major problems in the oil and gas industry[ The cylinder is the geometry most widely used in laboratory testing procedures for rocks and other geo!materials[ 3894
3895
Y[ Abousleiman and L[ Cui 1[ GOVERNING EQUATIONS
For the borehole and cylinder problems it is natural to use the cylindrical coordinate system to analyze the poromechanical behavior of such geometries exhibiting axial symmetry[ We limit the formulations and solutions to the transverse isotropic case\ keeping in mind that the extension to the orthotropic one "associated with the cylindrical coordinate system# is natural\ only its application is limited because it is necessary to determine the 02 poromechanical coe.cients needed[ In the case of transverse isotropy where only eight poromechanical parameters are involved with the z!axis assigned as the axis of material rotational symmetry\ stressÐstrain relations are presented below ] F srr J KM00 G G H Gsuu G HM01 G G H j szz f HM02 Jt F H 9 G ru G H G tuz G H 9 G G H f trz j k 9
M01
M02
9
9
M00
M02
9
9
M02
M22
9
9
9
9
M33
9
9
9
9
M44
9
9
9
9
9 LF orr J
FaJ HG G G G 9 HGouu G G a G HG G G G 9 Hj ozz f ja?f − p[ 9 HJgru F J 9 F HG G G G 9 HGguz G G 9 G HG G G G M44 lf grz j f 9 j
"0#
In the above\ oii and gij and sii and tij are strain components and total stress components in the cylindrical coordinate system\ respectively ^ p is the pore pressure\ a and a? are Biot|s e}ective stress coe.cients in the isotropic plane "rÐu plane# and in the z!direction\ respec! tively ^ and Mij are components of the drained elastic modulus matrix given by ] M00
M01
E"E?−En?1 # "0¦n#"E?−E?n−1En?1 # E"E?n¦En?1 # "0¦n#"E?−E?n−1En?1 #
M02
M22 M33 G
EE?n? E?−E?n−1En?1 E?1 "0−n# E?−E?n−1En?1
"1a#
"1b# "1c#
"1d#
E M00 −M01 1 1"0¦n#
"1e#
M44 G?
"1f#
in which E and n are drained Young|s modulus and Poisson|s ratio in the isotropic plane\ E? and n? are similar quantities related to the direction of the axis of symmetry\ and G? is the shear modulus related to the direction of the axis of symmetry[ The ~uid volumetric variation relation for the transversely isotropic material may be written as ]
z
p ¦aorr ¦aouu ¦a?ozz M
"2#
where z is the variation of ~uid content per unit reference volume\ and M is Biot|s modulus[ It is noted that unlike elasticity\ except for the permeability\ eight material properties are required to de_ne a transversely isotropic poroelastic material\ such as those mentioned above ] E\ E?\ n\ n?\ G?\ a\ a? and M[ However\ they need not necessarily be measured directly[ They can be determined from other parameters which are more convenient to measure in
Transversely isotropic media for wellbore and cylinder
3896
the laboratory[ For example\ the poroelastic properties under undrained conditions are probably easier to be measured[ A set of relations between the material properties adopted previously and the material properties related to the undrained state is listed below "see Cui\ 0884 ^ Cheng\ 0886# ]
a
M00 M u00 −a1 M
"3a#
M01 M u01 −a1 M
"3b#
M02 M u02 −aa?M
"3c#
M22 M u22 −a?1 M
"3d#
0 ðB"M u00 ¦M u01 #¦B?M u02 Ł 2M
"3e#
0 "1BM u02 ¦B?M u22 # 2M
"3f#
a?
where the superscript\ u\ indicates the undrained state ^ M uij are the elements of the undrained elastic matrix and can be expressed in terms of the undrained Young|s moduli and Poisson|s ratios by eqns "1# ^ B and B? are\ respectively\ Skempton|s coe.cients in the isotropic plane and in the direction of the axis of elastic symmetry[ The relation between variations of pore pressure and total stresses at the undrained state is de_ned through the Skempton|s coe.cients ] pu −02 "Bsurr ¦Bsuuu ¦B?suzz #[
"4#
It is assumed that the material is homogeneous\ the boundary conditions do not vary along the z!axis\ and the depth of the domain in the z!direction is in_nitely long or much greater than the cross!sectional geometry ^ hence\ the generalized plane strain "or complete plane strain# condition manifests itself "Abousleiman et al[\ 0885b ^ Cui et al[\ 0886b#[ The generalized plane strain condition results in that all stress and strain components\ and the pore pressure are z!independent ^ therefore\ leading to the following governing equations[ Equilibrium equations 1srr 0 1tur srr −suu ¦ ¦ 9 1r r 1u r
"5a#
1tru 0 1suu tru ¦ ¦1 9 1r r 1u r
"5b#
1trz 0 1tuz trz ¦ ¦ 9[ 1r r 1u r
"5c#
StrainÐdisplacement relation 1ur 1r
"6a#
0 1uu ur ¦ r 1u r
"6b#
1uz 1z
"6c#
0 1ur 1uu uu ¦ − r 1u 1r r
"6d#
orr ouu
ozz gru
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Y[ Abousleiman and L[ Cui
1uz 1r
"6e#
0 1uz r 1u
"6f#
grz guz
where ui are the components of the vector of displacement in the cylindrical coordinate system[ Continuity equation 1z −k91 p 9 1t
"7#
where k k:m is the permeability in the isotropic plane in which k is the intrinsic per! meability in the isotropic plane and m is the viscosity of the pore ~uid\ and 91 is a di}erential operator expressed by ]
91
11 1r
¦ 1
0 11 0 1 ¦ 1 1[ r 1r r 1u
"8#
From eqns "7#\ "5#\ "2#\ and "0#\ we obtained the following ~uid variation _eld equation ] 1z −cT 91 z 9 1t
"09#
where cT is a di}usion coe.cient in the isotropic plane and may be expressed by ]
cT
kMM00 M00 ¦a1 M
[
"00#
Darcy|s law 1p 1r
"01a#
0 1p r 1u
"01b#
qr −k qu −k
qz 9
"01c#
where qi are the components of the vector of the speci_c discharge of the pore ~uid in the cylindrical coordinate system[
2[ SOLUTION FOR BOREHOLE PROBLEM
2[0[ Problem description It is assumed that an in_nitely long borehole is drilled perpendicular to the isotropic plane of a transversely isotropic poroelastic formation[ The borehole axis is designated as
Transversely isotropic media for wellbore and cylinder
3898
Fig[ 0[ Inclined borehole in a transversely isotropic material[
the z!axis in Fig[ 0"a#[ The principal axes of the far!_eld stresses are shown as x?−y?−z? with three stress components denoted as Sx?\ Sy? and Sz?[ The inclination of the borehole is measured by the angles 8x and 8z as depicted in Fig[ 0"a#[ Figure 0"b# shows the far!_eld stress components from the vantage!point of the borehole coordinates x−y−z[ It is noted that there now exist normal and shear stress components denoted by Sij[ The boundary conditions of the problem can be imposed in the following ] at the far! _eld\ i[e[\ r : \ sxx −Sx
"02a#
syy −Sy
"02b#
szz −Sz
"02c#
txy −Sxy
"02d#
tyz −Syz
"02e#
txz −Sxz
"02f#
p p9
"02g#
where sii and tij are stress components under the Cartesian coordinate system xyz\ and p9 is the formation virgin pore pressure ^ and at the borehole wall\ i[e[\ r R srr −pw H"t#
"03a#
tru trz 9
"03b#
p pi H"t#
"03c#
in which pw is the wellbore pressure\ pi is the pore pressure at the borehole wall\ and H"t# is the Heaviside step function[ It is noted that pi is not necessary the same as pw due to the existence of a _lter cake that usually forms in drilling operations[ Because of the linearity of the problem\ it can be decomposed into three sub!problems[ Problem 0 At the far!_eld "r : #\
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Y[ Abousleiman and L[ Cui
szz −n?"Sx ¦Sy #−"a?−1n?a#p9
"04a#
sxx −Sx
"04b#
syy −Sy
"04c#
txy −Sxy
"04d#
p p9
"04e#
tyz txz 9 ^
"04f#
srr −pw H"t#
"05a#
tru 9
"05b#
trz 9
"05c#
p pi H"t#[
"05d#
szz −Sz ¦ðn?"Sx ¦Sy #¦"a?−1n?a#p9 Ł
"06a#
sxx syy txy tyz txz p 9 ^
"06b#
and\ at the borehole wall "r R#\
Problem II At the far!_eld "r : #\
and\ at the borehole wall "r R#\ srr tru trz p 9[
"07#
sxx syy szz txy p 9
"08a#
tyz −Syz
"08b#
txz −Sxz ^
"08c#
srr tru trz p 9[
"19#
Problem III At the far!_eld "r : #\
and\ at the borehole wall "r R#\
2[1[ Correspond elastic case Before moving to the poroelastic solution\ let us _rst examine the stresses around the borehole for the corresponding elastic problem[ This can be achieved by setting p p9 pi 9 in the poroelastic problem[ When the symmetry axis of the transversely isotropic material coincides with the borehole axis\ it can be demonstrated\ from the anisotropic elasticity "Amadei\ 0872#\ that the vertical stresses in Problem II and anti!plane shear stresses in Problem III "which are the only non!zero stress components# do not produce any in!plane stress[ Therefore\ they are totally uncoupled with Problem I[ More importantly\ Problem II does not disturb the initial stress _eld at all ^ and anti!shear stresses in Problem III have no e}ect on the volumetric strain[ This indicates that the original trivial pore pressure _elds in Problems II and III are never changed[ Therefore\ Problems II and III are stand!alone elastic problems\ and only Problem I is of a poroelastic nature[ In addition\ according to the anisotropic elastic borehole solution "Amadei\ 0872#\ the aniso! tropic elastic solution for stresses of Problem III is exactly the same as the isotropic one[
Transversely isotropic media for wellbore and cylinder
3800
However\ it should be noted that these conditions are no longer valid when the material symmetry axis has an arbitrary inclination[ Under this circumstance\ the pore pressure may be in~uenced by an anti!plane shear stress _eld[ 2[2[ Solution for Problem I Problem I is a plane strain problem[ In elasticity the plane strain solution for stresses in transverse isotropic material is the same as that for the isotropic one\ if the plane of symmetry is orthogonal to the material rotational axis of symmetry[ While the governing equations of the poroelastic anisotropic case are similar to the isotropic ones\ their solutions for stresses are di}erent "Abousleiman et al[\ 0884 ^ Cui\ 0884#[ The solution for Problem I is then expressed by ] srr"I# −P9 ¦S9 cos 1"u−ur #¦srr"0# ¦srr"1# ¦srr"2#
"10a#
"I# "0# "1# "2# suu −P9 −S9 cos 1"u−ur #¦suu ¦suu ¦suu
"10b#
"I# "I# szz n?ðsrr"I# ¦suu Ł−"a?−1n?a#p"I#
"10c#
"I# "2# tru −S9 sin 1"u−ur #¦tru
"10d#
"I# trz"I# tuz 9
"10e#
p"I# p9 ¦p"1# ¦p"2#
"10f#
where P9
S9
Sx ¦Sy 1
Sx −Sy 1 1 ¦S xy 1
X0
1
0 1Sxy [ ur tan−0 1 Sx −Sy
"11a#
"11b#
"11c#
"0# "1# "2# "2# Details of the derivations and expressions of srr"0# \ srr"1# \ srr"2# \ suu \ suu \ suu \ tru \ p"1#\ and p"2# are presented in Appendix A[
2[3[ Solution for Problem II As mentioned before\ Problem II is purely elastic for the case of material anisotropy concerned within this paper[ Its solution for stresses is the same as the one for the cor! responding isotropic problem "Cui et al[\ 0886a#\ hence\ "II# −Sz ¦ðn?"Sx ¦Sy #¦"a?−1n?a#p9 Ł szz
"12a#
"II# "II# "II# srr"II# suu tru trz"II# tuz p"II# 9[
"12b#
In fact\ this set of stresses and the pore pressure satis_es all the governing equations\ compatibility and boundary conditions ^ therefore\ it forms an admissible poroelastic solution[ Generally speaking\ taking o} the core material which occupied the borehole geometry originally disturbs initial stress and pore pressure _elds around the borehole and results in the redistribution of stresses and pore pressures[ Problem II presents the parts of stress and pore pressure _elds which are not disturbed through this process[ Therefore\ its solution must be uniform as the same as its initial state[
3801
Y[ Abousleiman and L[ Cui
2[4[ Solution for Problem III This problem is an anti!shear problem[ For this special case of transverse isotropy investigated here\ the shear deformation is uncoupled with the pore ~uid ~ow[ Since the anisotropic elasticity shows that the excavation of the borehole only disturbs the anti!shear stresses\ hence the shear deformation\ Problem III is also purely elastic[ The solution is the same as the elastic one[ Under these circumstances\ the anisotropy does not a}ect anti! shear stress _elds[ "However\ the displacements are a}ected[# Therefore\ the solution can be obtained as ]
$
trz"III# −"Sxz cos u¦Syz sin u# 0−H"t#
$
"III# "Sxz sin u−Syz cos u# 0¦H"t# tuz
R1 r1
R1 r1
%
"III# "III# "III# srr"III# suu szz tru p"III# 9[
%
"13a#
"13b# "13c#
2[5[ Final solution Finally\ the solution for dynamic variables of an inclined borehole in a transversely isotropic poroelastic medium is obtained by superposition ^ i[e[\ srr srr"I# suu s
"14a#
"I# uu
"I# zz
szz s ¦s
"14b# "II# zz
"I# tru tru
"14c# "14d#
"III# rz
"14e#
"III# uz
"14f#
trz t
tuz t
p p"I#
"14g#
in which components that are identically zero are omitted[ 3[ SOLUTION FOR CYLINDER PROBLEMS
3[0[ Problem description In this problem\ the length of the cylinder is assumed to be much greater than its radius ^ the isotropic plane of the material symmetry is also normal to the axis of the cylinder "the z!axis in Fig[ 1# ^ and\ the ~ow in the z!direction is not allowed[ The boundary
Fig[ 1[ Schematic of the cylinder[
Transversely isotropic media for wellbore and cylinder
3802
conditions commonly encountered in the laboratory can be summarized at the cylinder|s outer boundary\ r R\ as ] p p9 "t#
"15a#
qr Q"t#:1pR
"15b#
srr −P9 "t#−S9 cos uH"t#
"15c#
tru S9 sin 1uH"t#
"15d#
trz 9[
"15e#
In addition\ a uniform axial strain is imposed which is similar in laboratory tests jargon called a {{stroke control test|| ] i[e[\ ozz −o9 "t#[
"16#
In the above\ p9 is a time!dependent pore pressure applied on the cylindrical surface ^ Q"t# is the total ~ow discharge across the cylindrical surface per unit length ^ P9 is the normal pressure applied on the surface and it may be a time!dependent loading ^ S9 is a constant which represents the amplitude of sinusoidal distributions of normal pressure and shear stress applied on the surface ^ and o9 is the magnitude of the compressive axial strain and may also be time!dependent[ In view of the linearity\ the problem can be decomposed into the following modes for which combinations "modes superposition# can simulate many laboratory test con! _gurations ] Mode 0 At r R\ p p9 "t#
"17a#
srr 9
"17b#
tru 9
"17c#
qr Q"t#:1pR
"18a#
srr 9
"18b#
tru 9
"18c#
p 9
"29a#
srr −P9 "t#
"29b#
tru 9
"29c#
with the plane strain condition ozz 9[ Mode 1 At r R\
with the plane strain condition ozz 9[ Mode 2 At r R\
with the plane strain condition ozz 9[
3803
Y[ Abousleiman and L[ Cui
Mode 3 At r R\ qr 9
"20a#
srr −P9 "t#
"20b#
tru 9
"20c#
p 9
"21a#
srr −S9 cos 1uH"t#
"21b#
tru S9 sin 1uH"t#
"21c#
with the plane strain condition ozz 9[ Mode 4 At r R\
with the plane strain condition ozz 9[ Mode 5 At r R\ qr 9
"22a#
srr −S9 cos 1uH"t#
"22b#
tru S9 sin 1uH"t#
"22c#
with the plane strain condition ozz 9[ Mode 6 At r R\ p 9
"23a#
srr 9
"23b#
tru 9
"23c#
trz 9
"23d#
with a generalized plane strain condition ozz −o9[ Mode 7 At r R\ qr 9
"24a#
srr 9
"24b#
tru 9
"24c#
trz 9
"24d#
with a generalized plane strain condition ozz −o9[ Except for Modes 4 and 5\ which are asymmetric\ all other modes are axisymmetric[ Modes 4 and 5 are solved below _rst[ Then\ the general solution for the axisymmetric
Transversely isotropic media for wellbore and cylinder
3804
cylinder problem is derived[ Hence\ all axisymmetric modes are solved according to the corresponding boundary conditions[ 3[1[ Solution for asymmetric problem Modes 4 and 5 are asymmetric problems[ Their corresponding elastic solution can be obtained as ] srr −S9 cos 1uH"t#
"25a#
suu S9 cos 1uH"t#
"25b#
tru S9 sin 1uH"t#
"25c#
szz 9[
"25d#
It is noted that according to the formula "4#\ the stress _eld expressed by "25# does not result in any variation of the pore pressure at the undrained state\ and the ~ow boundary conditions for Modes 4 and 5 are trivial[ Hence\ the pore pressure distribution in the cylinder for Modes 4 and 5 must also be trivial since there is no disturbance subjected to the stress _eld "25#[ In other words\ Modes 4 and 5 are purely elastic[ Therefore\ the solution for Modes 4 and 5 are exactly the same as their elastic counterparts and expressed by ] srr −S9 cos 1uH"t#
"26a#
suu S9 cos 1uH"t#
"26b#
tru S9 sin 1uH"t#
"26c#
szz 9
"26d#
p 9[
"26e#
3[2[ Solution for axisymmetric problem For an axisymmetric cylinder problem\ such as all other modes except for Modes 4 and 5\ the axisymmetry leads to ] uu 9
"27a#
qu 9
"27b#
tru trz tuz 9[
"27c#
Except that uz is a function of r\ z\ and t\ all other variables\ such as srr\ suu\ szz\ p\ qr and ur\ are functions of the radial distance r and time t only[ The general solution for an axisymmetric cylinder problem can be derived analytically in the Laplace transform domain[ The solution in the time domain is obtained via the inverse of the Laplace transform through a numerical technique "Stehfest\ 0869#[ Because of axisymmetry\ the di}usion eqn "09# can be written as ] 11 z 0 1z 1z ¦ 9[ −cT 1t 1r1 r 1r
0
1
"28#
Applying the Laplace transform to eqn "28# yields ] s 0 dz½ − z½ 9 ¦ 1 r dr c T dr
d1 z½
"39#
where ½ denotes the Laplace transform and s is the Laplace transform variable[ Setting j rzs:cT \ it is obtained that ]
3805
Y[ Abousleiman and L[ Cui
aM M00 ¦a1 M
z½ A0 I9 "j#[
"30#
Applying the solution for the ~uid content variation z½ and using constitutive relations "0# and "2#\ the strainÐdisplacement relation "6#\ and Darcy|s law "01#\ the general solution for the axisymmetric cylinder problem "associated with Modes 0Ð3\ 6\ or 7# is derived as ] u½r p½
A0 zs:cT
I0 "j#¦A1 r
"31a#
M00 A0 I9 "j#−1aMA1 −a?Mo½zz a q½r −k
X
"31b#
s M00 A0 I0 "j# cT a
"31c#
I0 "j# M00 ¦M01 ¦1a1 M s½ rr M02 ¦aa?M −A0 ¦ A1 ¦ o½zz 1G j 1G 1G
"31d#
s½ uu I0 "j# M00 ¦M01 ¦1a1 M M02 ¦aa?M −A0 I9 "j#− A1 ¦ o½zz ¦ 1G j 1G 1G
"31e#
0
s½ zz −
0
1
1
a? M −M02 A0 I9 "j#¦1"M02 ¦aa?M#A1 ¦"M22 ¦a?1 M#o½zz [ a 00
"31f#
In the above\ A0 and A1 are arbitrary functions of the Laplace transform variable s ^ and In is the modi_ed Bessel function of the _rst kind of order n[ For individual problems\ A0 and A1 can be determined from corresponding loading and boundary conditions[ For Modes 0Ð3\ 6\ and 7\ A0 and A1 are obtained below ] Mode 0 o½zz 9 A0 p½9
"32a#
b M00 ¦M01 ¦1a1 M 1G C0 I0 "b#
"32b#
p½9 C0
"32c#
M00 ¦M01 ¦1a1 M M00 bI9 "b# −1aM[ 1G a I0 "b#
"33#
A1 where b Rzs:cT \ and C0 is expressed by ]
C0 Mode 1
A0 − A1 −
o½zz 9
"34a#
Q a 1pb kM00 I0 "b#
"34b#
1aG
Q 1
1pb kM00 "M00 ¦M01 ¦1a1 M#
[
"34c#
Transversely isotropic media for wellbore and cylinder
3806
Mode 2 o½zz 9
"35a#
9 a1 M P GC2 M00 I9 "b#
"35b#
9 P 1GC2
"35c#
A0 −
A1 − where C2 is expressed by ]
C2
M00 ¦M01 ¦1a1 M 1a1 M I0 "b# − [ 1G M00 bI9 "b#
"36#
Mode 3
A1 −
o½zz 9
"37a#
A0 9
"37b#
P 9 M00 ¦M01 ¦1a1 M
"37c#
[
Mode 6 o½zz −o½9 A0 −o½9
"38a#
0
0 C6 aa?M¦1a1 M M00 I9 "b# C2 A1 −o½9
C6 C2
1
"38b#
"38c#
where C2 is given by eqn "36# and C6 is expressed by ] C6
aa?M I0 "b# M02 ¦aa?M − [ M00 bI9 "b# 1G
"49#
Mode 7
A1 o½9
o½zz −o½9
"40a#
A0 9
"40b#
M02 ¦aa?M M00 ¦M01 ¦1a1 M
[
"40c#
It should be noted that A0 in Modes 3 and 7 is zero ^ hence\ no ~ow takes place in the cylinder[ In fact\ for Modes 3 and 7 the pore pressure is identical everywhere\ and the boundary surface is impermeable "jacketed test#[ Therefore\ the pore ~uid is not allowed to ~ow in the cylinder ^ and the modes represent the undrained state ^ i[e[\ solutions for Modes 3 and 7 are what we call the {{undrained solutions||[
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Y[ Abousleiman and L[ Cui 4[ NUMERICAL EXAMPLES
4[0[ Material anisotropy For transversely isotropic materials with micro!homogeneous and micro!isotropic solid skeletons\ the poroelastic constants "Biot|s e}ective stress parameter# a and a? can be expressed in terms of Mij and the bulk modulus of the solid grains\ Ks "Cheng\ 0886# ]
a 0−
M00 ¦M01 ¦M02 2Ks
"41a#
1M02 ¦M22 [ 2Ks
"41b#
a? 0−
Therefore\ Young|s moduli and Poisson|s ratios "which lead to all necessary Mij#\ M\ and Ks are enough to de_ne the transversely isotropic poroelastic constitutive relations[ As an example\ it is assumed that E 19[5 GPa\ n 9[078\ M 04[7 GPa\ Ks 37[1 GPa\ and k k? 7[53×09−5 m1:MPa day "Cui et al[\ 0885b#[ For given anisotropic ratios of nE E:E? and nn n:n?\ E? and n? can be determined ^ hence\ a transversely isotropic material is de_ned[ Di}erent ratios of nE and nn de_ne di}erent degrees of anisotropy\ and for nE nn 0 the material is isotropic[ In the following examples\ various ratios of E:E? and n:n? were chosen to demonstrate the e}ect of anisotropy[ 4[1[ Inclined borehole The far!_eld in situ stress and formation pore pressure gradients adopted for the analysis are the following "Woodland\ 0889# ] Sx? 18 MPa:km\ Sy? 19 MPa:km\ Sz? 14 MPa:km\ and p9 8[7 MPa:km[ The radius of the borehole was assumed to be R 9[0 m[ The deviation of the borehole was de_ned by 8x 29> and 8z 59> ^ and the depth of the formation was assumed to be 0999 m[ For the sake of convenience\ only the case of an excavation was analyzed ^ i[e[\ the well pressure was assumed to be zero[ Numerical results are presented in Figs 2Ð6[ In all the _gures\ the e}ective stress indicates Terzaghi|s e}ective stress "sij¦pdij# ^ and the negative values of stresses are pre! sented so that the results imply the rock mechanics sign convention for stresses "i[e[\ positive for compression#[ All calculated pore pressures and stresses are presented as varying with r:R along u 89>[ Figure 2 presents the pore pressure pro_les with nn 9[4\ 0\ 1 for _xed nE 0 at di}erent times[ It is observed that the material anisotropy in~uences the pore pressure only at small times "t 9[990 day#[ For a long time duration "t 0 day#\ all pore pressure distributions for di}erent nn are identical ^ indicating that Poisson|s ratios anisotropy does not a}ect pore pressure at large times[ In Fig[ 3 the e}ective radial stress distribution shows similar results of the anisotropy e}ect as for the pore pressure[ The stress for nn 9[4 is in~uenced much more by the anisotropy than for nn 1[ However\ unlike the case for the pressure\ the anisotropy e}ect on the stress still exists at large times[ It seems that the anisotropy e}ect on the e}ective radial stress becomes more signi_cant at places far away from the borehole wall as time progresses[ Figures 4 and 5 describe\ respectively\ the e}ective tangential and axial stresses around the borehole[ It can be seen from these _gures that at small time "t 9[990 day# for di}erent ratios of nn magnitudes of stresses around the borehole are quite di}erent\ but they are close at the places away from the borehole[ When time duration becomes longer "t 0 day#\ the anisotropy e}ect on stresses is transmitted to the places away from the borehole[ It should also be noted that the material anisotropy e}ect is signi_cant on e}ective tangential and axial stresses compared to the pore pressure and the radial stress ^ thus a}ecting wellbore stability analyses[ For the cases of varied nE with _xed nn 0\ analyses showed that material anisotropy e}ects on the pore pressure and all e}ective stresses are qualitatively similar to the previous
Transversely isotropic media for wellbore and cylinder
Fig[ 2[ For E:E? 0 and along the direction of u 89>\ pore pressure "p# near the borehole varying with r:R[
Fig[ 3[ For E:E? 0 and along the direction of u 89>\ e}ective radial stress "−s?rr# near the borehole varyuing with r:R[
3808
3819
Y[ Abousleiman and L[ Cui
Fig[ 4[ For E:E? 0 and along the direction of u 89>\ e}ective tangential stress "−s?uu# near the borehole varying with r:R[
Fig[ 5[ For E:E? 0 and along the direction of u 89>\ e}ective axial stress "−s?zz # near the borehole varying with r:R[
Transversely isotropic media for wellbore and cylinder
3810
Fig[ 6[ For n:n? 0 and along the direction of u 89>\ e}ective axial stress "−s?zz # near the borehole varying with r:R[
cases of varied nn[ However\ considerable e}ects of anisotropy are exhibited in the calcu! lation of the e}ective axial stresses as shown in Fig[ 6[ Similar to the cases of varied nn\ the material anisotropy related to the di}erence between E and E? in~uences the e}ective axial stress considerably[ Notice that material anisotropy would have a combination of these two ratios\ and the corresponding analysis should be conducted[ 4[2[ Cylinder The radius of the cylinder was set to 9[0 m[ It was assumed that a linearly time! dependent compressive axial strain rate "{{stroke rate||# of 09−7 0:s had been applied ^ i[e[\ ozz −09−7 t with a free stress at the cylindrical boundary\ srr tru trz p 9[ These conditions results in a uniaxial problem for an elastic medium ^ however\ a poroelastic cylinder behave pseudo!three!dimensional[ Figure 7 presents the pore pressure at the center of the cylinder as a function of time for the cases of nn 9[4\ 0\ 1 with nE 0[ It is observed that the pore pressure increases from the initial "trivial# level once the axial strain is applied[ After a certain time duration\ it keeps a constant level as time increases\ since the pore pressure _eld in the cylinder becomes steady[ The magnitude of the pore pressure changes signi_cantly with varied ratio of nn[ The greater the nn\ the higher the pore pressure[ In Fig[ 8 the pore pressure at the center of the cylinder is plotted for the cases of nE 9[4\ 0\ 1 with nn 0[ Similar phenomena to the previous ones are observed[ Figures 09 and 00 show the tangential stresses at the boundary surface "r R# for the cases of _xed nE and nn\ respectively[ The material anisotropy and time e}ects on the tangential stress are qualitatively similar to those on the pore pressure at the center[ It is known that for a non!porous elastic cylinder under the same boundary conditions the problem is uniaxial and the tangential stress is trivial[ However\ for the poroelastic problem a tensile tangential stress is developed as time increases[ This suggests that the saturated cylinder may be fracturing due to a compressive axial strain[ The total axial stress at the center varying with time for the cases of nE 9[4\ 0\ 1 with nn 0 is displayed in Fig[ 01[ In the _gure\ the negative values of the stress "i[e[\ positive
3811
Y[ Abousleiman and L[ Cui
Fig[ 7[ For E:E? 0\ pore presdsure "p# at r 9 in the cylinder varyuing with time[
Fig[ 8[ For n:n? 0\ pre pressure "p# at r 9 in the cylinder varying with time[
indicates a compressive stress# varying with time are plotted[ The axial stress is almost linearly dependent of time[ Material anisotropy e}ect on the stress is considerable ^ the magnitude of the stress increases as the ratio nE decreases[ It is noted that the magnitude
Transversely isotropic media for wellbore and cylinder
3812
Fig[ 09[ For E:E? 0\ tangential stress "suu # at r R in the cylinder varyuing with time[
Fig[ 00[ For n:n? 0\ tangential stress "suu # at r R in the cylinder varying with time[
of the axial stress is much greater than the ones of all other dynamic variables[ Considering that the slow axial strain rate and the free stress boundary conditions were applied\ these results are expected[ Although the results are not presented\ it is of interest to point out
3813
Y[ Abousleiman and L[ Cui
Fig[ 01[ For n:n? 0\ axial stress "−szz # at r 9 in the cylinder varying with time[
that the e}ect of variation of nn on the axial stress\ on the other hand\ is negligible for nE 0[ This indicates that the di}erence between two Poisson|s ratios may not be a major factor as material anisotropy to in~uence the axial stress[ In contrast with the axial stress\ the radial displacement at the boundary surface is sensitive to the variation of nn\ but insigni_cantly in~uenced by the change of nE[ The displacement varying with time for cases of nn 9[4\ 0\ 1 with nE 0 is presented in Fig[ 02[ It is observed that the displacement is also almost linearly dependent of time and a greater ratio nn leads to a larger displacement[ Finally\ unlike the elastic problem\ with coupled pore ~uid e}ect the cylinder problem under Mode 6 is not uniform any more[ A non!trivial pore pressure _eld results from the compaction of the cylinder ^ and this _eld is not uniform because of the zero pore pressure condition enforced at the boundary[ Hence\ the deformation in the axial direction is resisted by the pore ~uid and the resistance is also spatially!dependent[ This is basically why the poroelastic problem is pseudo!three!dimensional[ The radial and tangential stresses are thus generated in the poroelastic problem[ Figures 03 and 04 present the variations of the axial stress and pore pressure along the radial direction at t 2 min and t 29 min[ Non! uniform stress and pore pressure _elds are observed in the _gures[ Inhomogeneity of stress and pore pressure _elds is a}ected by the anisotropy of Poisson|s ratios[ If the strain rate is very small and the sample permeability is large\ the poroelastic e}ect may not be signi_cant ^ hence\ the anisotropy e}ect on the axial stress may not be very pronounced\ and could be equivalent to the corresponding elastic cases[ However\ if the loading rate is considerable and the permeability is small enough "Abousleiman et al[\ 0885b#\ the pore pressure generated will be signi_cant and may be comparable to the axial stress[ 5[ CONCLUSIONS AND DISCUSSIONS
For this special form of anisotropy in which the isotropic plane of the transversely isotropic material is perpendicular to the borehole axis and cylinder axis\ the poroelastic solutions for inclined boreholes and cylinders have been derived[ The borehole solution
Transversely isotropic media for wellbore and cylinder
Fig[ 02[ For E:E? 0\ radial displacement "ur # at r R in the cylinder varyuing with time[
Fig[ 03[ For E:E? 0\ axial stress "−szz# in the cylinder varying with r:R at di}erent times[
3814
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Y[ Abousleiman and L[ Cui
Fig[ 04[ For E:E? 0\ pore presdsure "p# in the cylinder varying with r:R at di}erent times[
may be applied to limited cases in practice[ However\ the cylinder solution can be applied to measurements of poroelastic properties of transversely isotropic materials\ as well as the strain recovery method for in situ stress determinations in transversely isotropic formations[ The analyses of an inclined borehole show that e}ective stresses and pore pressure around the borehole are in~uenced considerably by the material anisotropy\ especially when the Poisson|s ratio in the isotropic plane is smaller than the Poisson|s ratio related to the z!direction[ Particularly\ the material anisotropy e}ect on the e}ective tangential and axial stresses is signi_cant[ Therefore\ it is expected that the material anisotropy in~uences the borehole stability signi_cantly\ while the corresponding elastic solution predicts no anisotropy e}ect on the fracturing at all[ A cylinder subjected to an axial strain was also analyzed[ The results indicate that\ unlike the elastic cases\ the poroelastic cylinder is a pseudo!three!dimensional problem[ A pore pressure _eld\ as well as tangential and radial stress _elds\ are generated due to the compaction of the cylinder[ The material anisotropy e}ect on these _elds is signi_cant[ The analyses also show that if the axial strain rate is very small\ the magnitudes of generated pore pressure and in!plane stresses may not be signi_cant compared to the axial stress[ However\ for the materials with lower tensile strength\ such as soils and rocks\ the induced tensile stress may be high enough to fracture the specimen[ Acknowled`ements*The work is supported by a National Science Foundation grant to the Rock Mechanics Research Center\ the Oklahoma Center for the Advancement of Science and Technology\ and the O[U[ Rock Mechanics Consortium[ Discussions with Prof[ A[ H[!D[ Cheng of University of Delaware and Prof[ J[!C[ Roegiers of University of Oklahoma were bene_cial to the writing of this paper[
REFERENCES Abousleiman\ Y[\ Cui\ L[\ Cheng\ A[ H[!D[ and Roegiers\ J[!C[ "0884# Poroelastic solution of an inclined borehole in a transversely isotropic medium[ Proceedin`s of the 24th U[S[ Symposium on Rock Mechanics\ eds J[ J[ K[ Daemen and R[ A[ Schultz[ Abousleiman\ Y[\ Cheng\ A[ H[!D[\ Cui\ L[\ Detournay\ E[ and Roegiers\ J[!C[ "0885a# Mandel|s problem revisited[ Geotechnique 35"1#\ 076Ð084[
Transversely isotropic media for wellbore and cylinder
3816
Abousleiman\ Y[\ Cheng\ A[ H[!D[\ Jiang\ C[ and Roegiers\ J[!C[ "0885b# Poroviscoelastic analysis of borehole and cylinder problems[ Acta Mechanica 008\ 008Ð108[ Amadei\ B[ "0872# Rock Anisotropy and the Theory of Stress Measurements[ Springer!Verlag[ Biot\ M[ A[ "0844# Theory of elasticity and consolidation of a porous anisotropic solid[ Journal of Applied Physics 15\ 071Ð074[ Biot\ M[ A[ and Willis\ D[ G[ "0846# The elastic coe.cients of the theory of consolidation[ Journal of Applied Mechanics 13\ 483Ð590[ Cheng\ A[ H[!D[ "0886# Material coe.cient of anisotropic poroelasticity[ Int[ J[ Rock Mech[ Min[ Sci[ 23\ 088Ð 194[ Cui\ L[ "0884# Poroelasticity with application to rock mechanics[ Ph[D[ dissertation\ University of Delaware[ Cui\ L[\ Cheng\ A[ H[!D[ and Abousleiman\ Y[ "0886a# Poroelastic solution for an inclined borehole[ Journal of Applied Mechanics\ ASME 53\ 21Ð27[ Cui\ L[\ Abousleiman\ Y[\ Cheng\ A[ H[!D[ and Roegiers\ J[!C[ "0885a# Anisotropic e}ect on one!dimensional consolidation[ En`ineerin` Mechanics\ Vol[ 0\ Proceedin`s of the 00th En`ineerin` Mechanics Conference\ ASCE\ ed[ Y[ K[ Lin and T[ C[ Su\ pp[ 360Ð363[ Cui\ L[\ Abousleiman\ Y[\ Cheng\ A[ H[!D[\ Kaliakin\ V[ N[ and Roegiers\ J[!C[ "0885b# Finite element analysis of anisotropic poroelasticity ] a generalized Mandel|s problem and an inclined borehole problem[ Int[ J[ Numer[ Anal[ Methods Geomech[ 19\ 270Ð390[ Cui\ L[\ Kaliakin\ V[ N[\ Abousleiman\ Y[ and Cheng\ A[ H[!D[ "0886b# Finite element formulation and application of poroelastic generalized plane strain problems[ Int[ J[ Rock Mech[ Min[ Sci[ 23"5#\ 842Ð851[ Detournay\ E[ and Cheng\ A[ H[!D[ "0877# Poroelastic response of a borehole in a non!hydrostatic stress _eld[ Int[ J[ Rock Mech[ Min[ Sci[ Geomech[ and Abstr[ 14\ 060Ð071[ Rice\ J[ R[ and Clearly\ M[ P[ "0865# Some basic stress di}usion solutions for ~uid!saturated elastic porous media with compressible constituents[ Reviews of Geophysics and Space Physics 03"3#\ 116Ð130[ Stehfest\ H[ "0869# Numerical inversion of Laplace transforms[ Comm[ ACM[ 02\ 36Ð38 and 513[ Thompson\ M[ and Willis\ J[ R[ "0880# A reformulation of the equations of anisotropic poroelasticity[ Journal of Applied Mechanics\ ASME 47\ 501Ð505[ Woodland\ D[ C[ "0889# Borehole instability in the western Canadian overthrust belt[ SPE Drill[ En`[ 4\ 16Ð22[
APPENDIX In the solution for Problem I expressed by eqn "10#\ superscript {{i|| on the right!hand side indicates the corresponding solution for the ith mode which was described in Detournay and Cheng "0877#[ Actually\ Modes 0\ 1\ and 2 are three di}erent special plane strain borehole problems[ In all three modes\ the boundary conditions at in_nity "r : # are set as ] "i# p"i# 9[ srr"i# tru
"42#
However\ each mode has di}erent boundary conditions at the borehole wall[ More speci_cally ] Mode 0 At r R\ "0# p"0# 9[ srr"0# "P9 −pw #H"t# ^ tru
"43#
"1# srr"1# tru 9 ^ p"1# −"p9 −pi #H"t#[
"44#
"2# srr"2# −S9 cos 1"u−ur #H"t# ^ tru S9 sin 1"u−ur #H"t# ^ p"2# 9[
"45#
Mode 1 At r R\
Mode 2 At r R\
Similar to the isotropic problem\ the _rst two modes are axisymmetric problems[ The di}usion of pore pressure can thus be uncoupled from the solid deformation ^ and is governed by ] 1p 11 p 0 1p −cT ¦ 9[ 1t 1r1 r 1r
0
1
"46#
rp dr
"47#
It can be derived that the axisymmetry also leads to ] ur
a 0 A ¦ r M00 r
g
r
a
where A is an integration constant[ In the above\ all variables are _nite and o 9 at in_nity is implied[ Therefore\ the stresses can be expressed by ]
3817
Y[ Abousleiman and L[ Cui A
sr −1G
r su 1G
A r
1
1
0
¦a 0−
1 g
0
M01 0 M00 r1
1 g
r
−a 0− M01 0 M00 r1
r
rp dr
0
rp dr−a 0−
a
"48a#
a
1
M01 p[ M00
"48b#
Similar to the case of Problem III of the cylinder\ Mode 0 can be proven to be purely elastic since the variation of stresses around the borehole does not disturb the pore pressure _eld[ Hence\ the solution is obtained as ]
srr"0# H"t#"P9 −pw #
R1
"59a#
r1
"0# suu −H"t#"P9 −pw #
R1
"59b#
r1
p"0# 9[
"59c#
For Mode 1\ p"1# can be obtained from the di}usion eqn "46# ^ however\ it can be solved analytically only in "1# are given below as their Laplace transforms ] the Laplace transform domain[ p"1#\ srr"1# \ and suu p½"1# −
s½ rr"1# −
"1# s½ uu
p9 −pi K9 "j# s K9 "b#
R 1 K0 "b# p9 −pi 1Ga R K0 "j# − s M00 r bK9 "b# r1 bK9 "b#
$
"50a#
%
R 1 K0 "b# K9 "j# p9 −pi 1Ga R K0 "j# − ¦ 1 bK9 "b# s M00 r bK9 "b# K 9 "b# r
$
"50b#
%
"50c#
where Kn denotes the modi_ed Bessel function of the second kind of order n[ Again\ Mode 2 can only be solved analytically in the Laplace transform domain[ From the equilibrium eqns "5#\ the constitutive eqns "0# and "2#\ as well as the strainÐdisplacement relationship "6#\ the Navier|s equation in cylindrical coordinate system can be obtained as ] 0 1v 1p 1o −1G −a 9 1r r 1u 1r
"51a#
1v 0 1p 0 1o ¦1G −a 9 r 1u 1r r 1u
"51b#
M00 M00 where
v
0
1
0 0 1"ruu # 0 1ur − [ 1 r 1r r 1u
"52#
Applying the Laplace transform to Navier|s eqns "51a#\ and the di}usion eqn "09# yields ] aM 1z½ M00 ¦a1 M 1o½ 0 1v ½ − − 9 1G 1r r 1u 1G 1r
"53a#
½ M00 ¦a1 M 0 1o½ 1v aM 0 1z½ ¦ − 9 1G r 1u 1r 1G r 1u
"53b#
11 z½ 1r
1
¦
0 11 z½ 0 1z½ s ¦ − z½ 9[ r 1r r1 1u1 cT
"53c#
In view of the asymmetry of Mode 2\ it can be assumed that ] Sr \ Su \ P "z½ "2# \ o½ "2# \ u½r"2# \ s½ r"2# \ s½ u"2# \ p½"2# # "Z \ U \ E r\ # cos 1"u−ur #
"54a#
"2# ½ "2# \ u½u"2# \ t½ ru # "W Sru # sin 1"u−ur #[ "v \U u\
"54b#
Substituting relations "54# into eqns "53#\ a set of ordinary di}erential equations is produced[ They are ]
Transversely isotropic media for wellbore and cylinder
3818
W aM dZ M00 ¦a1 M dE −1 − 9 1G dr r 1G dr
"55a#
aM Z M00 ¦a1 M E 0 dW − − 9 1G r 1 dr 1G r
"55b#
d1 Z dr1
¦
0
1
0 dZ s 1 − r ¦3 Z 9[ r dr cT
"55c#
Solving the above set of ordinary di}erential eqns "55# with associated boundary conditions\ the following expressions are obtained ] S9 cT R1 C0 K1 "j#¦A0 C1 s 1Gk r1
$
p½"2# cos 1"u−ur #
s½ rr"2# cos 1"u−ur #
%
"56a#
S9 5 0 R1 R3 A0 C0 K0 "j#¦ K1 "j# −A1 C1 −2C2 s j j1 r1 r3
$ 0
"2# cos 1"u−ur # s½ uu
1
S9 5 0 R3 −A0 C0 K0 "j#¦ 0¦ K1 "j# ¦2C2 1 s j j r3
6
$
0
"2# sin 1"u−ur # t½ ru
¦
2 1
j
1
K1 "j# −
$
1
0
S9 0 1A0 C0 K0 "j# s j
A1 R1 R3 C1 −2C2 1 1 r r3
%
%
7
%
"56b#
"56c#
"56d#
"56e#
where C0
3 1A0 "M2 −M1 #−A1 M0 3M0 1A0 "M2 −M1 #−A1 M0
"57b#
1A0 "M1 ¦M2 #¦2A1 M0 [ 2ð1A0 "M2 −M1 #−A1 M0 Ł
"57c#
C1 − C2
"57a#
In the above\ A0
A1
aM
"58a#
M00 ¦a1 M
M00 ¦M01 ¦1a1 M
"58b#
M00 ¦a1 M
M0
M00 K "b# 1Ga 1
"58c#
0 5 M1 K0 "b#¦ K1 "b# b b1 M2 1
0
"58d#
1
2 0 K "b#¦ K1 "b# [ b 0 b1
"58e#
"1# "2# "2# \ suu \ tru \ p"1#\ and p"2# in the time domain\ a numerical technique for the inverse of the To evaluate srr"1# \ srr"2# \ suu Laplace transform should be applied[
Int[ J[ Solids Structures Vol[ 24\ Nos 23Ð24\ pp[ 3894Ð3818\ 0887 Þ 0887 Elsevier Science Ltd[ All rights reserved Printed in Great Britain 9919Ð6572:87:,*see front matter
Pergamon
PII ] S9919Ð6572"87#99090Ð1
POROELASTIC SOLUTIONS IN TRANSVERSELY ISOTROPIC MEDIA FOR WELLBORE AND CYLINDER Y[ ABOUSLEIMAN Department of Civil Engineering\ School of Engineering and Architecture\ The Lebanese American University\ Byblos\ Lebanon
and L[ CUI Rock Mechanics Institute\ The University of Oklahoma\ Norman\ Oklahoma 62908\ U[S[A[ "Received 0 September 0886 ^ in revised form 01 February 0887# Abstract*This paper presents closed!form solutions for the pore pressures and stress _elds for the inclined borehole and the cylinder\ induced by boundary stress perturbation in an anisotropic poroelastic medium[ The governing equations for the transversely isotropic poroelastic materials under the cylindrical coordinate system are presented[ The solution for the inclined borehole\ subjected to a three!dimensional far!_eld anisotropic stress _eld\ is derived with the borehole generator coinciding with the material axis of symmetry ^ while the solutions for the cylinder are obtained under various loading conditions that are encountered in laboratory testing procedures[ The analysis shows\ in addition to the e}ect of time on stress and pore pressure variations\ that poromechanical anisotropic material coe.cients also play an important role in calculating the in! plane stress _elds\ which is not the case using the classical anisotropic theory of elasticity[ Þ 0887 Elsevier Science Ltd[ All rights reserved[
0[ INTRODUCTION
The theory of anisotropic poroelasticity was formulated by Biot "0844#\ in an attempt to generalize consolidation theory[ Later\ Biot and Willis "0846# extended the theory to describe various poromechanical properties of an anisotropic ~uid saturated medium[ In their work\ the outlined experimental procedures for the determination of the directional poromechanical properties are extremely di.cult to achieve\ if not impossible[ Following the interpretation of Rice and Cleary "0865# of the poroelastic material constants\ Thomp! son and Willis "0880# reformatted the anisotropic poroelastic constitutive relations\ in which the parameters that appear are given explicitly in terms of drained "elastic# and undrained compliances\ and the generalized tensors of Skempton|s pore pressure coe.cient\ Bij\ and Biot|s e}ective stress coe.cient\ aij[ Recently\ these parameters have been cast in a practical model for laboratory measurement "Cui\ 0884 ^ Cheng\ 0886#[ With these latest developments in material characterizations\ the application of the anisotropic poroelastic theory to engineering problems becomes more amenable[ Although geomaterials exhibit many forms of anisotropy\ closed!form solutions for anisotropic por! oelasticity are almost non!existent in the literature[ Recently\ the transverse anisotropic poroelastic solutions for the one!dimensional consolidation "Cui et al[\ 0885a# and the two! dimensional Mandel|s problem "Abousleiman et al[\ 0885a# were derived[ These solutions may not _nd a wide range of engineering applications\ but they can serve as benchmarks validating numerical algorithms such as the _nite element method "Cui et al[\ 0885b#[ In this work we extend the transverse isotropic poroelastic solutions to the cases of the inclined borehole and cylinder geometries[ Stability of wellbores in deep drilling remains one of the major problems in the oil and gas industry[ The cylinder is the geometry most widely used in laboratory testing procedures for rocks and other geo!materials[ 3894
3895
Y[ Abousleiman and L[ Cui 1[ GOVERNING EQUATIONS
For the borehole and cylinder problems it is natural to use the cylindrical coordinate system to analyze the poromechanical behavior of such geometries exhibiting axial symmetry[ We limit the formulations and solutions to the transverse isotropic case\ keeping in mind that the extension to the orthotropic one "associated with the cylindrical coordinate system# is natural\ only its application is limited because it is necessary to determine the 02 poromechanical coe.cients needed[ In the case of transverse isotropy where only eight poromechanical parameters are involved with the z!axis assigned as the axis of material rotational symmetry\ stressÐstrain relations are presented below ] F srr J KM00 G G H Gsuu G HM01 G G H j szz f HM02 Jt F H 9 G ru G H G tuz G H 9 G G H f trz j k 9
M01
M02
9
9
M00
M02
9
9
M02
M22
9
9
9
9
M33
9
9
9
9
M44
9
9
9
9
9 LF orr J
FaJ HG G G G 9 HGouu G G a G HG G G G 9 Hj ozz f ja?f − p[ 9 HJgru F J 9 F HG G G G 9 HGguz G G 9 G HG G G G M44 lf grz j f 9 j
"0#
In the above\ oii and gij and sii and tij are strain components and total stress components in the cylindrical coordinate system\ respectively ^ p is the pore pressure\ a and a? are Biot|s e}ective stress coe.cients in the isotropic plane "rÐu plane# and in the z!direction\ respec! tively ^ and Mij are components of the drained elastic modulus matrix given by ] M00
M01
E"E?−En?1 # "0¦n#"E?−E?n−1En?1 # E"E?n¦En?1 # "0¦n#"E?−E?n−1En?1 #
M02
M22 M33 G
EE?n? E?−E?n−1En?1 E?1 "0−n# E?−E?n−1En?1
"1a#
"1b# "1c#
"1d#
E M00 −M01 1 1"0¦n#
"1e#
M44 G?
"1f#
in which E and n are drained Young|s modulus and Poisson|s ratio in the isotropic plane\ E? and n? are similar quantities related to the direction of the axis of symmetry\ and G? is the shear modulus related to the direction of the axis of symmetry[ The ~uid volumetric variation relation for the transversely isotropic material may be written as ]
z
p ¦aorr ¦aouu ¦a?ozz M
"2#
where z is the variation of ~uid content per unit reference volume\ and M is Biot|s modulus[ It is noted that unlike elasticity\ except for the permeability\ eight material properties are required to de_ne a transversely isotropic poroelastic material\ such as those mentioned above ] E\ E?\ n\ n?\ G?\ a\ a? and M[ However\ they need not necessarily be measured directly[ They can be determined from other parameters which are more convenient to measure in
Transversely isotropic media for wellbore and cylinder
3896
the laboratory[ For example\ the poroelastic properties under undrained conditions are probably easier to be measured[ A set of relations between the material properties adopted previously and the material properties related to the undrained state is listed below "see Cui\ 0884 ^ Cheng\ 0886# ]
a
M00 M u00 −a1 M
"3a#
M01 M u01 −a1 M
"3b#
M02 M u02 −aa?M
"3c#
M22 M u22 −a?1 M
"3d#
0 ðB"M u00 ¦M u01 #¦B?M u02 Ł 2M
"3e#
0 "1BM u02 ¦B?M u22 # 2M
"3f#
a?
where the superscript\ u\ indicates the undrained state ^ M uij are the elements of the undrained elastic matrix and can be expressed in terms of the undrained Young|s moduli and Poisson|s ratios by eqns "1# ^ B and B? are\ respectively\ Skempton|s coe.cients in the isotropic plane and in the direction of the axis of elastic symmetry[ The relation between variations of pore pressure and total stresses at the undrained state is de_ned through the Skempton|s coe.cients ] pu −02 "Bsurr ¦Bsuuu ¦B?suzz #[
"4#
It is assumed that the material is homogeneous\ the boundary conditions do not vary along the z!axis\ and the depth of the domain in the z!direction is in_nitely long or much greater than the cross!sectional geometry ^ hence\ the generalized plane strain "or complete plane strain# condition manifests itself "Abousleiman et al[\ 0885b ^ Cui et al[\ 0886b#[ The generalized plane strain condition results in that all stress and strain components\ and the pore pressure are z!independent ^ therefore\ leading to the following governing equations[ Equilibrium equations 1srr 0 1tur srr −suu ¦ ¦ 9 1r r 1u r
"5a#
1tru 0 1suu tru ¦ ¦1 9 1r r 1u r
"5b#
1trz 0 1tuz trz ¦ ¦ 9[ 1r r 1u r
"5c#
StrainÐdisplacement relation 1ur 1r
"6a#
0 1uu ur ¦ r 1u r
"6b#
1uz 1z
"6c#
0 1ur 1uu uu ¦ − r 1u 1r r
"6d#
orr ouu
ozz gru
3897
Y[ Abousleiman and L[ Cui
1uz 1r
"6e#
0 1uz r 1u
"6f#
grz guz
where ui are the components of the vector of displacement in the cylindrical coordinate system[ Continuity equation 1z −k91 p 9 1t
"7#
where k k:m is the permeability in the isotropic plane in which k is the intrinsic per! meability in the isotropic plane and m is the viscosity of the pore ~uid\ and 91 is a di}erential operator expressed by ]
91
11 1r
¦ 1
0 11 0 1 ¦ 1 1[ r 1r r 1u
"8#
From eqns "7#\ "5#\ "2#\ and "0#\ we obtained the following ~uid variation _eld equation ] 1z −cT 91 z 9 1t
"09#
where cT is a di}usion coe.cient in the isotropic plane and may be expressed by ]
cT
kMM00 M00 ¦a1 M
[
"00#
Darcy|s law 1p 1r
"01a#
0 1p r 1u
"01b#
qr −k qu −k
qz 9
"01c#
where qi are the components of the vector of the speci_c discharge of the pore ~uid in the cylindrical coordinate system[
2[ SOLUTION FOR BOREHOLE PROBLEM
2[0[ Problem description It is assumed that an in_nitely long borehole is drilled perpendicular to the isotropic plane of a transversely isotropic poroelastic formation[ The borehole axis is designated as
Transversely isotropic media for wellbore and cylinder
3898
Fig[ 0[ Inclined borehole in a transversely isotropic material[
the z!axis in Fig[ 0"a#[ The principal axes of the far!_eld stresses are shown as x?−y?−z? with three stress components denoted as Sx?\ Sy? and Sz?[ The inclination of the borehole is measured by the angles 8x and 8z as depicted in Fig[ 0"a#[ Figure 0"b# shows the far!_eld stress components from the vantage!point of the borehole coordinates x−y−z[ It is noted that there now exist normal and shear stress components denoted by Sij[ The boundary conditions of the problem can be imposed in the following ] at the far! _eld\ i[e[\ r : \ sxx −Sx
"02a#
syy −Sy
"02b#
szz −Sz
"02c#
txy −Sxy
"02d#
tyz −Syz
"02e#
txz −Sxz
"02f#
p p9
"02g#
where sii and tij are stress components under the Cartesian coordinate system xyz\ and p9 is the formation virgin pore pressure ^ and at the borehole wall\ i[e[\ r R srr −pw H"t#
"03a#
tru trz 9
"03b#
p pi H"t#
"03c#
in which pw is the wellbore pressure\ pi is the pore pressure at the borehole wall\ and H"t# is the Heaviside step function[ It is noted that pi is not necessary the same as pw due to the existence of a _lter cake that usually forms in drilling operations[ Because of the linearity of the problem\ it can be decomposed into three sub!problems[ Problem 0 At the far!_eld "r : #\
3809
Y[ Abousleiman and L[ Cui
szz −n?"Sx ¦Sy #−"a?−1n?a#p9
"04a#
sxx −Sx
"04b#
syy −Sy
"04c#
txy −Sxy
"04d#
p p9
"04e#
tyz txz 9 ^
"04f#
srr −pw H"t#
"05a#
tru 9
"05b#
trz 9
"05c#
p pi H"t#[
"05d#
szz −Sz ¦ðn?"Sx ¦Sy #¦"a?−1n?a#p9 Ł
"06a#
sxx syy txy tyz txz p 9 ^
"06b#
and\ at the borehole wall "r R#\
Problem II At the far!_eld "r : #\
and\ at the borehole wall "r R#\ srr tru trz p 9[
"07#
sxx syy szz txy p 9
"08a#
tyz −Syz
"08b#
txz −Sxz ^
"08c#
srr tru trz p 9[
"19#
Problem III At the far!_eld "r : #\
and\ at the borehole wall "r R#\
2[1[ Correspond elastic case Before moving to the poroelastic solution\ let us _rst examine the stresses around the borehole for the corresponding elastic problem[ This can be achieved by setting p p9 pi 9 in the poroelastic problem[ When the symmetry axis of the transversely isotropic material coincides with the borehole axis\ it can be demonstrated\ from the anisotropic elasticity "Amadei\ 0872#\ that the vertical stresses in Problem II and anti!plane shear stresses in Problem III "which are the only non!zero stress components# do not produce any in!plane stress[ Therefore\ they are totally uncoupled with Problem I[ More importantly\ Problem II does not disturb the initial stress _eld at all ^ and anti!shear stresses in Problem III have no e}ect on the volumetric strain[ This indicates that the original trivial pore pressure _elds in Problems II and III are never changed[ Therefore\ Problems II and III are stand!alone elastic problems\ and only Problem I is of a poroelastic nature[ In addition\ according to the anisotropic elastic borehole solution "Amadei\ 0872#\ the aniso! tropic elastic solution for stresses of Problem III is exactly the same as the isotropic one[
Transversely isotropic media for wellbore and cylinder
3800
However\ it should be noted that these conditions are no longer valid when the material symmetry axis has an arbitrary inclination[ Under this circumstance\ the pore pressure may be in~uenced by an anti!plane shear stress _eld[ 2[2[ Solution for Problem I Problem I is a plane strain problem[ In elasticity the plane strain solution for stresses in transverse isotropic material is the same as that for the isotropic one\ if the plane of symmetry is orthogonal to the material rotational axis of symmetry[ While the governing equations of the poroelastic anisotropic case are similar to the isotropic ones\ their solutions for stresses are di}erent "Abousleiman et al[\ 0884 ^ Cui\ 0884#[ The solution for Problem I is then expressed by ] srr"I# −P9 ¦S9 cos 1"u−ur #¦srr"0# ¦srr"1# ¦srr"2#
"10a#
"I# "0# "1# "2# suu −P9 −S9 cos 1"u−ur #¦suu ¦suu ¦suu
"10b#
"I# "I# szz n?ðsrr"I# ¦suu Ł−"a?−1n?a#p"I#
"10c#
"I# "2# tru −S9 sin 1"u−ur #¦tru
"10d#
"I# trz"I# tuz 9
"10e#
p"I# p9 ¦p"1# ¦p"2#
"10f#
where P9
S9
Sx ¦Sy 1
Sx −Sy 1 1 ¦S xy 1
X0
1
0 1Sxy [ ur tan−0 1 Sx −Sy
"11a#
"11b#
"11c#
"0# "1# "2# "2# Details of the derivations and expressions of srr"0# \ srr"1# \ srr"2# \ suu \ suu \ suu \ tru \ p"1#\ and p"2# are presented in Appendix A[
2[3[ Solution for Problem II As mentioned before\ Problem II is purely elastic for the case of material anisotropy concerned within this paper[ Its solution for stresses is the same as the one for the cor! responding isotropic problem "Cui et al[\ 0886a#\ hence\ "II# −Sz ¦ðn?"Sx ¦Sy #¦"a?−1n?a#p9 Ł szz
"12a#
"II# "II# "II# srr"II# suu tru trz"II# tuz p"II# 9[
"12b#
In fact\ this set of stresses and the pore pressure satis_es all the governing equations\ compatibility and boundary conditions ^ therefore\ it forms an admissible poroelastic solution[ Generally speaking\ taking o} the core material which occupied the borehole geometry originally disturbs initial stress and pore pressure _elds around the borehole and results in the redistribution of stresses and pore pressures[ Problem II presents the parts of stress and pore pressure _elds which are not disturbed through this process[ Therefore\ its solution must be uniform as the same as its initial state[
3801
Y[ Abousleiman and L[ Cui
2[4[ Solution for Problem III This problem is an anti!shear problem[ For this special case of transverse isotropy investigated here\ the shear deformation is uncoupled with the pore ~uid ~ow[ Since the anisotropic elasticity shows that the excavation of the borehole only disturbs the anti!shear stresses\ hence the shear deformation\ Problem III is also purely elastic[ The solution is the same as the elastic one[ Under these circumstances\ the anisotropy does not a}ect anti! shear stress _elds[ "However\ the displacements are a}ected[# Therefore\ the solution can be obtained as ]
$
trz"III# −"Sxz cos u¦Syz sin u# 0−H"t#
$
"III# "Sxz sin u−Syz cos u# 0¦H"t# tuz
R1 r1
R1 r1
%
"III# "III# "III# srr"III# suu szz tru p"III# 9[
%
"13a#
"13b# "13c#
2[5[ Final solution Finally\ the solution for dynamic variables of an inclined borehole in a transversely isotropic poroelastic medium is obtained by superposition ^ i[e[\ srr srr"I# suu s
"14a#
"I# uu
"I# zz
szz s ¦s
"14b# "II# zz
"I# tru tru
"14c# "14d#
"III# rz
"14e#
"III# uz
"14f#
trz t
tuz t
p p"I#
"14g#
in which components that are identically zero are omitted[ 3[ SOLUTION FOR CYLINDER PROBLEMS
3[0[ Problem description In this problem\ the length of the cylinder is assumed to be much greater than its radius ^ the isotropic plane of the material symmetry is also normal to the axis of the cylinder "the z!axis in Fig[ 1# ^ and\ the ~ow in the z!direction is not allowed[ The boundary
Fig[ 1[ Schematic of the cylinder[
Transversely isotropic media for wellbore and cylinder
3802
conditions commonly encountered in the laboratory can be summarized at the cylinder|s outer boundary\ r R\ as ] p p9 "t#
"15a#
qr Q"t#:1pR
"15b#
srr −P9 "t#−S9 cos uH"t#
"15c#
tru S9 sin 1uH"t#
"15d#
trz 9[
"15e#
In addition\ a uniform axial strain is imposed which is similar in laboratory tests jargon called a {{stroke control test|| ] i[e[\ ozz −o9 "t#[
"16#
In the above\ p9 is a time!dependent pore pressure applied on the cylindrical surface ^ Q"t# is the total ~ow discharge across the cylindrical surface per unit length ^ P9 is the normal pressure applied on the surface and it may be a time!dependent loading ^ S9 is a constant which represents the amplitude of sinusoidal distributions of normal pressure and shear stress applied on the surface ^ and o9 is the magnitude of the compressive axial strain and may also be time!dependent[ In view of the linearity\ the problem can be decomposed into the following modes for which combinations "modes superposition# can simulate many laboratory test con! _gurations ] Mode 0 At r R\ p p9 "t#
"17a#
srr 9
"17b#
tru 9
"17c#
qr Q"t#:1pR
"18a#
srr 9
"18b#
tru 9
"18c#
p 9
"29a#
srr −P9 "t#
"29b#
tru 9
"29c#
with the plane strain condition ozz 9[ Mode 1 At r R\
with the plane strain condition ozz 9[ Mode 2 At r R\
with the plane strain condition ozz 9[
3803
Y[ Abousleiman and L[ Cui
Mode 3 At r R\ qr 9
"20a#
srr −P9 "t#
"20b#
tru 9
"20c#
p 9
"21a#
srr −S9 cos 1uH"t#
"21b#
tru S9 sin 1uH"t#
"21c#
with the plane strain condition ozz 9[ Mode 4 At r R\
with the plane strain condition ozz 9[ Mode 5 At r R\ qr 9
"22a#
srr −S9 cos 1uH"t#
"22b#
tru S9 sin 1uH"t#
"22c#
with the plane strain condition ozz 9[ Mode 6 At r R\ p 9
"23a#
srr 9
"23b#
tru 9
"23c#
trz 9
"23d#
with a generalized plane strain condition ozz −o9[ Mode 7 At r R\ qr 9
"24a#
srr 9
"24b#
tru 9
"24c#
trz 9
"24d#
with a generalized plane strain condition ozz −o9[ Except for Modes 4 and 5\ which are asymmetric\ all other modes are axisymmetric[ Modes 4 and 5 are solved below _rst[ Then\ the general solution for the axisymmetric
Transversely isotropic media for wellbore and cylinder
3804
cylinder problem is derived[ Hence\ all axisymmetric modes are solved according to the corresponding boundary conditions[ 3[1[ Solution for asymmetric problem Modes 4 and 5 are asymmetric problems[ Their corresponding elastic solution can be obtained as ] srr −S9 cos 1uH"t#
"25a#
suu S9 cos 1uH"t#
"25b#
tru S9 sin 1uH"t#
"25c#
szz 9[
"25d#
It is noted that according to the formula "4#\ the stress _eld expressed by "25# does not result in any variation of the pore pressure at the undrained state\ and the ~ow boundary conditions for Modes 4 and 5 are trivial[ Hence\ the pore pressure distribution in the cylinder for Modes 4 and 5 must also be trivial since there is no disturbance subjected to the stress _eld "25#[ In other words\ Modes 4 and 5 are purely elastic[ Therefore\ the solution for Modes 4 and 5 are exactly the same as their elastic counterparts and expressed by ] srr −S9 cos 1uH"t#
"26a#
suu S9 cos 1uH"t#
"26b#
tru S9 sin 1uH"t#
"26c#
szz 9
"26d#
p 9[
"26e#
3[2[ Solution for axisymmetric problem For an axisymmetric cylinder problem\ such as all other modes except for Modes 4 and 5\ the axisymmetry leads to ] uu 9
"27a#
qu 9
"27b#
tru trz tuz 9[
"27c#
Except that uz is a function of r\ z\ and t\ all other variables\ such as srr\ suu\ szz\ p\ qr and ur\ are functions of the radial distance r and time t only[ The general solution for an axisymmetric cylinder problem can be derived analytically in the Laplace transform domain[ The solution in the time domain is obtained via the inverse of the Laplace transform through a numerical technique "Stehfest\ 0869#[ Because of axisymmetry\ the di}usion eqn "09# can be written as ] 11 z 0 1z 1z ¦ 9[ −cT 1t 1r1 r 1r
0
1
"28#
Applying the Laplace transform to eqn "28# yields ] s 0 dz½ − z½ 9 ¦ 1 r dr c T dr
d1 z½
"39#
where ½ denotes the Laplace transform and s is the Laplace transform variable[ Setting j rzs:cT \ it is obtained that ]
3805
Y[ Abousleiman and L[ Cui
aM M00 ¦a1 M
z½ A0 I9 "j#[
"30#
Applying the solution for the ~uid content variation z½ and using constitutive relations "0# and "2#\ the strainÐdisplacement relation "6#\ and Darcy|s law "01#\ the general solution for the axisymmetric cylinder problem "associated with Modes 0Ð3\ 6\ or 7# is derived as ] u½r p½
A0 zs:cT
I0 "j#¦A1 r
"31a#
M00 A0 I9 "j#−1aMA1 −a?Mo½zz a q½r −k
X
"31b#
s M00 A0 I0 "j# cT a
"31c#
I0 "j# M00 ¦M01 ¦1a1 M s½ rr M02 ¦aa?M −A0 ¦ A1 ¦ o½zz 1G j 1G 1G
"31d#
s½ uu I0 "j# M00 ¦M01 ¦1a1 M M02 ¦aa?M −A0 I9 "j#− A1 ¦ o½zz ¦ 1G j 1G 1G
"31e#
0
s½ zz −
0
1
1
a? M −M02 A0 I9 "j#¦1"M02 ¦aa?M#A1 ¦"M22 ¦a?1 M#o½zz [ a 00
"31f#
In the above\ A0 and A1 are arbitrary functions of the Laplace transform variable s ^ and In is the modi_ed Bessel function of the _rst kind of order n[ For individual problems\ A0 and A1 can be determined from corresponding loading and boundary conditions[ For Modes 0Ð3\ 6\ and 7\ A0 and A1 are obtained below ] Mode 0 o½zz 9 A0 p½9
"32a#
b M00 ¦M01 ¦1a1 M 1G C0 I0 "b#
"32b#
p½9 C0
"32c#
M00 ¦M01 ¦1a1 M M00 bI9 "b# −1aM[ 1G a I0 "b#
"33#
A1 where b Rzs:cT \ and C0 is expressed by ]
C0 Mode 1
A0 − A1 −
o½zz 9
"34a#
Q a 1pb kM00 I0 "b#
"34b#
1aG
Q 1
1pb kM00 "M00 ¦M01 ¦1a1 M#
[
"34c#
Transversely isotropic media for wellbore and cylinder
3806
Mode 2 o½zz 9
"35a#
9 a1 M P GC2 M00 I9 "b#
"35b#
9 P 1GC2
"35c#
A0 −
A1 − where C2 is expressed by ]
C2
M00 ¦M01 ¦1a1 M 1a1 M I0 "b# − [ 1G M00 bI9 "b#
"36#
Mode 3
A1 −
o½zz 9
"37a#
A0 9
"37b#
P 9 M00 ¦M01 ¦1a1 M
"37c#
[
Mode 6 o½zz −o½9 A0 −o½9
"38a#
0
0 C6 aa?M¦1a1 M M00 I9 "b# C2 A1 −o½9
C6 C2
1
"38b#
"38c#
where C2 is given by eqn "36# and C6 is expressed by ] C6
aa?M I0 "b# M02 ¦aa?M − [ M00 bI9 "b# 1G
"49#
Mode 7
A1 o½9
o½zz −o½9
"40a#
A0 9
"40b#
M02 ¦aa?M M00 ¦M01 ¦1a1 M
[
"40c#
It should be noted that A0 in Modes 3 and 7 is zero ^ hence\ no ~ow takes place in the cylinder[ In fact\ for Modes 3 and 7 the pore pressure is identical everywhere\ and the boundary surface is impermeable "jacketed test#[ Therefore\ the pore ~uid is not allowed to ~ow in the cylinder ^ and the modes represent the undrained state ^ i[e[\ solutions for Modes 3 and 7 are what we call the {{undrained solutions||[
3807
Y[ Abousleiman and L[ Cui 4[ NUMERICAL EXAMPLES
4[0[ Material anisotropy For transversely isotropic materials with micro!homogeneous and micro!isotropic solid skeletons\ the poroelastic constants "Biot|s e}ective stress parameter# a and a? can be expressed in terms of Mij and the bulk modulus of the solid grains\ Ks "Cheng\ 0886# ]
a 0−
M00 ¦M01 ¦M02 2Ks
"41a#
1M02 ¦M22 [ 2Ks
"41b#
a? 0−
Therefore\ Young|s moduli and Poisson|s ratios "which lead to all necessary Mij#\ M\ and Ks are enough to de_ne the transversely isotropic poroelastic constitutive relations[ As an example\ it is assumed that E 19[5 GPa\ n 9[078\ M 04[7 GPa\ Ks 37[1 GPa\ and k k? 7[53×09−5 m1:MPa day "Cui et al[\ 0885b#[ For given anisotropic ratios of nE E:E? and nn n:n?\ E? and n? can be determined ^ hence\ a transversely isotropic material is de_ned[ Di}erent ratios of nE and nn de_ne di}erent degrees of anisotropy\ and for nE nn 0 the material is isotropic[ In the following examples\ various ratios of E:E? and n:n? were chosen to demonstrate the e}ect of anisotropy[ 4[1[ Inclined borehole The far!_eld in situ stress and formation pore pressure gradients adopted for the analysis are the following "Woodland\ 0889# ] Sx? 18 MPa:km\ Sy? 19 MPa:km\ Sz? 14 MPa:km\ and p9 8[7 MPa:km[ The radius of the borehole was assumed to be R 9[0 m[ The deviation of the borehole was de_ned by 8x 29> and 8z 59> ^ and the depth of the formation was assumed to be 0999 m[ For the sake of convenience\ only the case of an excavation was analyzed ^ i[e[\ the well pressure was assumed to be zero[ Numerical results are presented in Figs 2Ð6[ In all the _gures\ the e}ective stress indicates Terzaghi|s e}ective stress "sij¦pdij# ^ and the negative values of stresses are pre! sented so that the results imply the rock mechanics sign convention for stresses "i[e[\ positive for compression#[ All calculated pore pressures and stresses are presented as varying with r:R along u 89>[ Figure 2 presents the pore pressure pro_les with nn 9[4\ 0\ 1 for _xed nE 0 at di}erent times[ It is observed that the material anisotropy in~uences the pore pressure only at small times "t 9[990 day#[ For a long time duration "t 0 day#\ all pore pressure distributions for di}erent nn are identical ^ indicating that Poisson|s ratios anisotropy does not a}ect pore pressure at large times[ In Fig[ 3 the e}ective radial stress distribution shows similar results of the anisotropy e}ect as for the pore pressure[ The stress for nn 9[4 is in~uenced much more by the anisotropy than for nn 1[ However\ unlike the case for the pressure\ the anisotropy e}ect on the stress still exists at large times[ It seems that the anisotropy e}ect on the e}ective radial stress becomes more signi_cant at places far away from the borehole wall as time progresses[ Figures 4 and 5 describe\ respectively\ the e}ective tangential and axial stresses around the borehole[ It can be seen from these _gures that at small time "t 9[990 day# for di}erent ratios of nn magnitudes of stresses around the borehole are quite di}erent\ but they are close at the places away from the borehole[ When time duration becomes longer "t 0 day#\ the anisotropy e}ect on stresses is transmitted to the places away from the borehole[ It should also be noted that the material anisotropy e}ect is signi_cant on e}ective tangential and axial stresses compared to the pore pressure and the radial stress ^ thus a}ecting wellbore stability analyses[ For the cases of varied nE with _xed nn 0\ analyses showed that material anisotropy e}ects on the pore pressure and all e}ective stresses are qualitatively similar to the previous
Transversely isotropic media for wellbore and cylinder
Fig[ 2[ For E:E? 0 and along the direction of u 89>\ pore pressure "p# near the borehole varying with r:R[
Fig[ 3[ For E:E? 0 and along the direction of u 89>\ e}ective radial stress "−s?rr# near the borehole varyuing with r:R[
3808
3819
Y[ Abousleiman and L[ Cui
Fig[ 4[ For E:E? 0 and along the direction of u 89>\ e}ective tangential stress "−s?uu# near the borehole varying with r:R[
Fig[ 5[ For E:E? 0 and along the direction of u 89>\ e}ective axial stress "−s?zz # near the borehole varying with r:R[
Transversely isotropic media for wellbore and cylinder
3810
Fig[ 6[ For n:n? 0 and along the direction of u 89>\ e}ective axial stress "−s?zz # near the borehole varying with r:R[
cases of varied nn[ However\ considerable e}ects of anisotropy are exhibited in the calcu! lation of the e}ective axial stresses as shown in Fig[ 6[ Similar to the cases of varied nn\ the material anisotropy related to the di}erence between E and E? in~uences the e}ective axial stress considerably[ Notice that material anisotropy would have a combination of these two ratios\ and the corresponding analysis should be conducted[ 4[2[ Cylinder The radius of the cylinder was set to 9[0 m[ It was assumed that a linearly time! dependent compressive axial strain rate "{{stroke rate||# of 09−7 0:s had been applied ^ i[e[\ ozz −09−7 t with a free stress at the cylindrical boundary\ srr tru trz p 9[ These conditions results in a uniaxial problem for an elastic medium ^ however\ a poroelastic cylinder behave pseudo!three!dimensional[ Figure 7 presents the pore pressure at the center of the cylinder as a function of time for the cases of nn 9[4\ 0\ 1 with nE 0[ It is observed that the pore pressure increases from the initial "trivial# level once the axial strain is applied[ After a certain time duration\ it keeps a constant level as time increases\ since the pore pressure _eld in the cylinder becomes steady[ The magnitude of the pore pressure changes signi_cantly with varied ratio of nn[ The greater the nn\ the higher the pore pressure[ In Fig[ 8 the pore pressure at the center of the cylinder is plotted for the cases of nE 9[4\ 0\ 1 with nn 0[ Similar phenomena to the previous ones are observed[ Figures 09 and 00 show the tangential stresses at the boundary surface "r R# for the cases of _xed nE and nn\ respectively[ The material anisotropy and time e}ects on the tangential stress are qualitatively similar to those on the pore pressure at the center[ It is known that for a non!porous elastic cylinder under the same boundary conditions the problem is uniaxial and the tangential stress is trivial[ However\ for the poroelastic problem a tensile tangential stress is developed as time increases[ This suggests that the saturated cylinder may be fracturing due to a compressive axial strain[ The total axial stress at the center varying with time for the cases of nE 9[4\ 0\ 1 with nn 0 is displayed in Fig[ 01[ In the _gure\ the negative values of the stress "i[e[\ positive
3811
Y[ Abousleiman and L[ Cui
Fig[ 7[ For E:E? 0\ pore presdsure "p# at r 9 in the cylinder varyuing with time[
Fig[ 8[ For n:n? 0\ pre pressure "p# at r 9 in the cylinder varying with time[
indicates a compressive stress# varying with time are plotted[ The axial stress is almost linearly dependent of time[ Material anisotropy e}ect on the stress is considerable ^ the magnitude of the stress increases as the ratio nE decreases[ It is noted that the magnitude
Transversely isotropic media for wellbore and cylinder
3812
Fig[ 09[ For E:E? 0\ tangential stress "suu # at r R in the cylinder varyuing with time[
Fig[ 00[ For n:n? 0\ tangential stress "suu # at r R in the cylinder varying with time[
of the axial stress is much greater than the ones of all other dynamic variables[ Considering that the slow axial strain rate and the free stress boundary conditions were applied\ these results are expected[ Although the results are not presented\ it is of interest to point out
3813
Y[ Abousleiman and L[ Cui
Fig[ 01[ For n:n? 0\ axial stress "−szz # at r 9 in the cylinder varying with time[
that the e}ect of variation of nn on the axial stress\ on the other hand\ is negligible for nE 0[ This indicates that the di}erence between two Poisson|s ratios may not be a major factor as material anisotropy to in~uence the axial stress[ In contrast with the axial stress\ the radial displacement at the boundary surface is sensitive to the variation of nn\ but insigni_cantly in~uenced by the change of nE[ The displacement varying with time for cases of nn 9[4\ 0\ 1 with nE 0 is presented in Fig[ 02[ It is observed that the displacement is also almost linearly dependent of time and a greater ratio nn leads to a larger displacement[ Finally\ unlike the elastic problem\ with coupled pore ~uid e}ect the cylinder problem under Mode 6 is not uniform any more[ A non!trivial pore pressure _eld results from the compaction of the cylinder ^ and this _eld is not uniform because of the zero pore pressure condition enforced at the boundary[ Hence\ the deformation in the axial direction is resisted by the pore ~uid and the resistance is also spatially!dependent[ This is basically why the poroelastic problem is pseudo!three!dimensional[ The radial and tangential stresses are thus generated in the poroelastic problem[ Figures 03 and 04 present the variations of the axial stress and pore pressure along the radial direction at t 2 min and t 29 min[ Non! uniform stress and pore pressure _elds are observed in the _gures[ Inhomogeneity of stress and pore pressure _elds is a}ected by the anisotropy of Poisson|s ratios[ If the strain rate is very small and the sample permeability is large\ the poroelastic e}ect may not be signi_cant ^ hence\ the anisotropy e}ect on the axial stress may not be very pronounced\ and could be equivalent to the corresponding elastic cases[ However\ if the loading rate is considerable and the permeability is small enough "Abousleiman et al[\ 0885b#\ the pore pressure generated will be signi_cant and may be comparable to the axial stress[ 5[ CONCLUSIONS AND DISCUSSIONS
For this special form of anisotropy in which the isotropic plane of the transversely isotropic material is perpendicular to the borehole axis and cylinder axis\ the poroelastic solutions for inclined boreholes and cylinders have been derived[ The borehole solution
Transversely isotropic media for wellbore and cylinder
Fig[ 02[ For E:E? 0\ radial displacement "ur # at r R in the cylinder varyuing with time[
Fig[ 03[ For E:E? 0\ axial stress "−szz# in the cylinder varying with r:R at di}erent times[
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Y[ Abousleiman and L[ Cui
Fig[ 04[ For E:E? 0\ pore presdsure "p# in the cylinder varying with r:R at di}erent times[
may be applied to limited cases in practice[ However\ the cylinder solution can be applied to measurements of poroelastic properties of transversely isotropic materials\ as well as the strain recovery method for in situ stress determinations in transversely isotropic formations[ The analyses of an inclined borehole show that e}ective stresses and pore pressure around the borehole are in~uenced considerably by the material anisotropy\ especially when the Poisson|s ratio in the isotropic plane is smaller than the Poisson|s ratio related to the z!direction[ Particularly\ the material anisotropy e}ect on the e}ective tangential and axial stresses is signi_cant[ Therefore\ it is expected that the material anisotropy in~uences the borehole stability signi_cantly\ while the corresponding elastic solution predicts no anisotropy e}ect on the fracturing at all[ A cylinder subjected to an axial strain was also analyzed[ The results indicate that\ unlike the elastic cases\ the poroelastic cylinder is a pseudo!three!dimensional problem[ A pore pressure _eld\ as well as tangential and radial stress _elds\ are generated due to the compaction of the cylinder[ The material anisotropy e}ect on these _elds is signi_cant[ The analyses also show that if the axial strain rate is very small\ the magnitudes of generated pore pressure and in!plane stresses may not be signi_cant compared to the axial stress[ However\ for the materials with lower tensile strength\ such as soils and rocks\ the induced tensile stress may be high enough to fracture the specimen[ Acknowled`ements*The work is supported by a National Science Foundation grant to the Rock Mechanics Research Center\ the Oklahoma Center for the Advancement of Science and Technology\ and the O[U[ Rock Mechanics Consortium[ Discussions with Prof[ A[ H[!D[ Cheng of University of Delaware and Prof[ J[!C[ Roegiers of University of Oklahoma were bene_cial to the writing of this paper[
REFERENCES Abousleiman\ Y[\ Cui\ L[\ Cheng\ A[ H[!D[ and Roegiers\ J[!C[ "0884# Poroelastic solution of an inclined borehole in a transversely isotropic medium[ Proceedin`s of the 24th U[S[ Symposium on Rock Mechanics\ eds J[ J[ K[ Daemen and R[ A[ Schultz[ Abousleiman\ Y[\ Cheng\ A[ H[!D[\ Cui\ L[\ Detournay\ E[ and Roegiers\ J[!C[ "0885a# Mandel|s problem revisited[ Geotechnique 35"1#\ 076Ð084[
Transversely isotropic media for wellbore and cylinder
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Abousleiman\ Y[\ Cheng\ A[ H[!D[\ Jiang\ C[ and Roegiers\ J[!C[ "0885b# Poroviscoelastic analysis of borehole and cylinder problems[ Acta Mechanica 008\ 008Ð108[ Amadei\ B[ "0872# Rock Anisotropy and the Theory of Stress Measurements[ Springer!Verlag[ Biot\ M[ A[ "0844# Theory of elasticity and consolidation of a porous anisotropic solid[ Journal of Applied Physics 15\ 071Ð074[ Biot\ M[ A[ and Willis\ D[ G[ "0846# The elastic coe.cients of the theory of consolidation[ Journal of Applied Mechanics 13\ 483Ð590[ Cheng\ A[ H[!D[ "0886# Material coe.cient of anisotropic poroelasticity[ Int[ J[ Rock Mech[ Min[ Sci[ 23\ 088Ð 194[ Cui\ L[ "0884# Poroelasticity with application to rock mechanics[ Ph[D[ dissertation\ University of Delaware[ Cui\ L[\ Cheng\ A[ H[!D[ and Abousleiman\ Y[ "0886a# Poroelastic solution for an inclined borehole[ Journal of Applied Mechanics\ ASME 53\ 21Ð27[ Cui\ L[\ Abousleiman\ Y[\ Cheng\ A[ H[!D[ and Roegiers\ J[!C[ "0885a# Anisotropic e}ect on one!dimensional consolidation[ En`ineerin` Mechanics\ Vol[ 0\ Proceedin`s of the 00th En`ineerin` Mechanics Conference\ ASCE\ ed[ Y[ K[ Lin and T[ C[ Su\ pp[ 360Ð363[ Cui\ L[\ Abousleiman\ Y[\ Cheng\ A[ H[!D[\ Kaliakin\ V[ N[ and Roegiers\ J[!C[ "0885b# Finite element analysis of anisotropic poroelasticity ] a generalized Mandel|s problem and an inclined borehole problem[ Int[ J[ Numer[ Anal[ Methods Geomech[ 19\ 270Ð390[ Cui\ L[\ Kaliakin\ V[ N[\ Abousleiman\ Y[ and Cheng\ A[ H[!D[ "0886b# Finite element formulation and application of poroelastic generalized plane strain problems[ Int[ J[ Rock Mech[ Min[ Sci[ 23"5#\ 842Ð851[ Detournay\ E[ and Cheng\ A[ H[!D[ "0877# Poroelastic response of a borehole in a non!hydrostatic stress _eld[ Int[ J[ Rock Mech[ Min[ Sci[ Geomech[ and Abstr[ 14\ 060Ð071[ Rice\ J[ R[ and Clearly\ M[ P[ "0865# Some basic stress di}usion solutions for ~uid!saturated elastic porous media with compressible constituents[ Reviews of Geophysics and Space Physics 03"3#\ 116Ð130[ Stehfest\ H[ "0869# Numerical inversion of Laplace transforms[ Comm[ ACM[ 02\ 36Ð38 and 513[ Thompson\ M[ and Willis\ J[ R[ "0880# A reformulation of the equations of anisotropic poroelasticity[ Journal of Applied Mechanics\ ASME 47\ 501Ð505[ Woodland\ D[ C[ "0889# Borehole instability in the western Canadian overthrust belt[ SPE Drill[ En`[ 4\ 16Ð22[
APPENDIX In the solution for Problem I expressed by eqn "10#\ superscript {{i|| on the right!hand side indicates the corresponding solution for the ith mode which was described in Detournay and Cheng "0877#[ Actually\ Modes 0\ 1\ and 2 are three di}erent special plane strain borehole problems[ In all three modes\ the boundary conditions at in_nity "r : # are set as ] "i# p"i# 9[ srr"i# tru
"42#
However\ each mode has di}erent boundary conditions at the borehole wall[ More speci_cally ] Mode 0 At r R\ "0# p"0# 9[ srr"0# "P9 −pw #H"t# ^ tru
"43#
"1# srr"1# tru 9 ^ p"1# −"p9 −pi #H"t#[
"44#
"2# srr"2# −S9 cos 1"u−ur #H"t# ^ tru S9 sin 1"u−ur #H"t# ^ p"2# 9[
"45#
Mode 1 At r R\
Mode 2 At r R\
Similar to the isotropic problem\ the _rst two modes are axisymmetric problems[ The di}usion of pore pressure can thus be uncoupled from the solid deformation ^ and is governed by ] 1p 11 p 0 1p −cT ¦ 9[ 1t 1r1 r 1r
0
1
"46#
rp dr
"47#
It can be derived that the axisymmetry also leads to ] ur
a 0 A ¦ r M00 r
g
r
a
where A is an integration constant[ In the above\ all variables are _nite and o 9 at in_nity is implied[ Therefore\ the stresses can be expressed by ]
3817
Y[ Abousleiman and L[ Cui A
sr −1G
r su 1G
A r
1
1
0
¦a 0−
1 g
0
M01 0 M00 r1
1 g
r
−a 0− M01 0 M00 r1
r
rp dr
0
rp dr−a 0−
a
"48a#
a
1
M01 p[ M00
"48b#
Similar to the case of Problem III of the cylinder\ Mode 0 can be proven to be purely elastic since the variation of stresses around the borehole does not disturb the pore pressure _eld[ Hence\ the solution is obtained as ]
srr"0# H"t#"P9 −pw #
R1
"59a#
r1
"0# suu −H"t#"P9 −pw #
R1
"59b#
r1
p"0# 9[
"59c#
For Mode 1\ p"1# can be obtained from the di}usion eqn "46# ^ however\ it can be solved analytically only in "1# are given below as their Laplace transforms ] the Laplace transform domain[ p"1#\ srr"1# \ and suu p½"1# −
s½ rr"1# −
"1# s½ uu
p9 −pi K9 "j# s K9 "b#
R 1 K0 "b# p9 −pi 1Ga R K0 "j# − s M00 r bK9 "b# r1 bK9 "b#
$
"50a#
%
R 1 K0 "b# K9 "j# p9 −pi 1Ga R K0 "j# − ¦ 1 bK9 "b# s M00 r bK9 "b# K 9 "b# r
$
"50b#
%
"50c#
where Kn denotes the modi_ed Bessel function of the second kind of order n[ Again\ Mode 2 can only be solved analytically in the Laplace transform domain[ From the equilibrium eqns "5#\ the constitutive eqns "0# and "2#\ as well as the strainÐdisplacement relationship "6#\ the Navier|s equation in cylindrical coordinate system can be obtained as ] 0 1v 1p 1o −1G −a 9 1r r 1u 1r
"51a#
1v 0 1p 0 1o ¦1G −a 9 r 1u 1r r 1u
"51b#
M00 M00 where
v
0
1
0 0 1"ruu # 0 1ur − [ 1 r 1r r 1u
"52#
Applying the Laplace transform to Navier|s eqns "51a#\ and the di}usion eqn "09# yields ] aM 1z½ M00 ¦a1 M 1o½ 0 1v ½ − − 9 1G 1r r 1u 1G 1r
"53a#
½ M00 ¦a1 M 0 1o½ 1v aM 0 1z½ ¦ − 9 1G r 1u 1r 1G r 1u
"53b#
11 z½ 1r
1
¦
0 11 z½ 0 1z½ s ¦ − z½ 9[ r 1r r1 1u1 cT
"53c#
In view of the asymmetry of Mode 2\ it can be assumed that ] Sr \ Su \ P "z½ "2# \ o½ "2# \ u½r"2# \ s½ r"2# \ s½ u"2# \ p½"2# # "Z \ U \ E r\ # cos 1"u−ur #
"54a#
"2# ½ "2# \ u½u"2# \ t½ ru # "W Sru # sin 1"u−ur #[ "v \U u\
"54b#
Substituting relations "54# into eqns "53#\ a set of ordinary di}erential equations is produced[ They are ]
Transversely isotropic media for wellbore and cylinder
3818
W aM dZ M00 ¦a1 M dE −1 − 9 1G dr r 1G dr
"55a#
aM Z M00 ¦a1 M E 0 dW − − 9 1G r 1 dr 1G r
"55b#
d1 Z dr1
¦
0
1
0 dZ s 1 − r ¦3 Z 9[ r dr cT
"55c#
Solving the above set of ordinary di}erential eqns "55# with associated boundary conditions\ the following expressions are obtained ] S9 cT R1 C0 K1 "j#¦A0 C1 s 1Gk r1
$
p½"2# cos 1"u−ur #
s½ rr"2# cos 1"u−ur #
%
"56a#
S9 5 0 R1 R3 A0 C0 K0 "j#¦ K1 "j# −A1 C1 −2C2 s j j1 r1 r3
$ 0
"2# cos 1"u−ur # s½ uu
1
S9 5 0 R3 −A0 C0 K0 "j#¦ 0¦ K1 "j# ¦2C2 1 s j j r3
6
$
0
"2# sin 1"u−ur # t½ ru
¦
2 1
j
1
K1 "j# −
$
1
0
S9 0 1A0 C0 K0 "j# s j
A1 R1 R3 C1 −2C2 1 1 r r3
%
%
7
%
"56b#
"56c#
"56d#
"56e#
where C0
3 1A0 "M2 −M1 #−A1 M0 3M0 1A0 "M2 −M1 #−A1 M0
"57b#
1A0 "M1 ¦M2 #¦2A1 M0 [ 2ð1A0 "M2 −M1 #−A1 M0 Ł
"57c#
C1 − C2
"57a#
In the above\ A0
A1
aM
"58a#
M00 ¦a1 M
M00 ¦M01 ¦1a1 M
"58b#
M00 ¦a1 M
M0
M00 K "b# 1Ga 1
"58c#
0 5 M1 K0 "b#¦ K1 "b# b b1 M2 1
0
"58d#
1
2 0 K "b#¦ K1 "b# [ b 0 b1
"58e#
"1# "2# "2# \ suu \ tru \ p"1#\ and p"2# in the time domain\ a numerical technique for the inverse of the To evaluate srr"1# \ srr"2# \ suu Laplace transform should be applied[