.
I
Societv of PetroleumEnaineers I
SPE 30752
Coupled Fluid Flow and Geomechanics 1. Theory and Governing Equations
in Reservoir Study -
H.-Y. Chen, SPE, L.W. Teufel, SPE, and R.L. Lee, SPE, New Mexico Instituteof Mining and Technology
Copyright 1995, Society of Petroleum Engineers, Inc. Tlus paper was prepsred for presentation at the SPE Annual Technical Conference &
Exhlbmon held in Dallas, U S.A,
22-25 October, 1995. by an SPE Program committee following rewew 01 of the paper, as reflect any
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m an abstract submitted The material,
by the author(s). Contents
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acknowledgment Box 833836,
of where and by whom the paper is presented. Wrrfe Librarian, SPE, U S A, fax 01-214-952-9435
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Abstraot The purpose of this study is to examine Biot’s two-phase (fluid and rock), isothermal, linear poroelastic theory from the conventional porous fluid-flow modeling point of view. Biot’s theory and the published applications are oriented more toward rock mechanics than fluid flow. Our goal is to preserve the commonly used systematic porous fluid-flow modeling and include geomechanics as an additional module. By developing such an approach, complex reservoir situations involving geomechanical issues (e.g., naturally fractured reservoirs, stress-sensitive reservoirs) can be pursued more systematically and easily. We show how the conventional fluid-flow formulations is extended to a coupled fluid-flow-geomechanics model. Consistent interpretation of various rock compressibilities and the effective stress law are shown to be critical in achieving the coupling. The “total (or system) compressibility” commonly used in reservoir engineering is shown to be a function of boundary conditions. qL. .:--1 . ..”4 . ...”. /:”a&A-:. l. fi . . ..a- . . . . . -O+a.-i”l 11-4.. UllUC1 UK MI1+UG>L LAKX (l>ULIU~lb llU111U&G11GUU3 111~LG11c4J monerties). satisfies a fourth-order equation r–-–– ~-_r–– .––,, the fluid nressure instead of the conventional second-order diffusion equation. Limiting cases include nondeformable, incompressible fluid and soiici, and constant mean normai stress are analyzed.
\
Introduction All petroleum reservoir problems involve two basic elements: fluid and rock. We are interested in two particular processes associated with them: fluid flow and geomechanics. Fluid flow is essential in a petroleum reservoir study. Geomechanics is believed to be important in the study of naturally fractured reservoirs and in reservoirs exhibiting stress-sensitivity. Wila lll>L T& theory describing fhUid-SOiki LWupllllg ‘--’-’: -“-”-‘--’ presented in a series papers by Biot. 1-7 Biot’s theory and the published applications are oriented more toward rock mechanics than fluid flow. Extension of Biot’s theory to reservoir studies is not straightforward, especifll y to nongeomechanical or fluid-flow oriented engineers. The purpose of this paper is to describe how the conventional fluid-flow modeling can be extended to a modeling. geomechanical fluid-flow and coupled Identification of the linkages and consistent interpretations between the flow and deformation fields are emphasized. Several excellent reviews or re-interpretations of Biot’s poroelasticity have been presented in, e.g., Refs. 8 through 15. Among these references, the works by Verruijt10 and Bear15 are the two most pertinent to this study. They also considered porous fluid-flow modeling approach coupled with Biot’s theory. Both works, however, assumed incompressible solid phase in both flow and deformation fields. The solid phase compressibility, although not the primal mechanism in rock deformations, is a necessity for a complete interpretation of Biot’s theory. Specificallyy, the solid compressibility is implicit in the so-called effective stress coefficient which, as will be shown, is the most iiiipMtrtt ad ~riiid ~@K@ ill tiie t}le~~~d jNiWiSStiCiiy”. ~-~ ~~I~rn@Qn Qf’ —_---illCOfllprcSSihie solid ~hafe effeCtiVei~ eliminates the consideration of effective stress coefficient and greatly simplifies the problem. We include the solid .. .. . compresslblllt y in both fiow and ciefonnation fieids to
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COUPLED FLUID FLOW AND GEOMECHANICS IN RESERVOIR STUDY
SPE 30752
hiohlioht -6--5--
tha --
imnnrtanm= -..yw. —..
nf “.
thr= . .
.-ffaAv. . . . . . .. . .
ctram .=. ””
.naffi.innt .V . . . . ..”ll.
w.alme it,, T.almAx,ath,arnfi,rkn
v W,vw.., , “,a., v“ trm .“ &uu .,,” “u~
D“,
cnl;A -. .Ln.xm h., E,T 2 ‘2 /A,, a au *u” Vvu “y J+. \ubLu
and the essentials of linear poroelastic theory. A comparison with Biot’s results then can be made as complete as possible. The commonly used notations in the petroleum industry will be honored in most of the developments. Stress and Strfi me nositive =__–.–._ in tensiQn where~s “prMsu#’ (~.g,, fll~d pressure) is positive for compression. Fluid-Flow Theory Equations governing isothermal, single-phase fluid flow in a deformable porous medium are derived in this section, The derivations naturally lead to geomechanical issues which will be incorporated in the section “Linear Poroelastic Theory.” Basic Relations. The three basic principles of fluid flow in porous media are: mass conservation, Darcy’s law, and equation of state. Mathematical y, these are: 10’ 15)16 Mass conservation:
~y~~:
Gersevanov in 1934 as cited by Biot2 and VerruijtlO). The condition for a nondefonnable (stationary) medium is v~=O. Gravity effects are not considered. Fluid density and viscosity are assumed to be a function nf fhirl a.n.a anl., .,a-d.m+ n Em A ..” ha kta-ta A “1 .Jwu +w. p...w” “-J . 17A. . “, ““-w. L, tiy. -r Ua.u Uu Iu.u&aLwu
to
to give p=poexp[c(’p-po)] where p. and PO are reference density and reference pressure, respectively. Rock properties are a function of mean normal stress and fluid pressure. The compressibility of the rock phase will be introduced later. Governing Equation. Eq. 1 gives Introducing Darcy’s law (Eq. 3) into
v
or v
()
pG’p
=Q&+vs
.V(($p)+($pv.vs,
P
............(5)
~.(p$v)
,
(Ih ~(+p) =(), .............................(1) at
()
u ,-_
p~vp
,
=y++pv.
-.
v,,
. . . . . . . . . . . . . . . . . . . . . . .,.........(6)
Solid: Darcy’s law:
V“[p, (l–+)vsl+
a[(l-$)psl = O, ........(2)
at
where d(.)/dt is the materiaJ derivative with respect to a moving solid defined as ,-l{\
Ur)
F– , ............................................(3)
a(.h W( ) —+ at
VS. V(-). .................................................(7)
+(V-V,)=-:VP
Equation of state (isothermal fluid compressibility): JaP P . ............................................................(4)
Eq. 7 links the material derivative, which is a Lagrangian concept, to a spatial or Eulerian description, Note that for nondefonnable media the materkd derivative is equivalent to a partial derivative since vS=O. Expanding the right-hand-side of Eq. 6 results in
ap
In Eqs. 1 through 4, p is the fluid density (mass per unit fluid volume), v is the fluid velocity vector, c is the fluid compressibility, and the subscripts refers to the “solid phase” counterparts. Also, V and V. denote gradient and divergence, respectively, p is the fluid viscosity, k is the permeability, $ is the eflective porosity, p is the fluid pressure, and t is time. rmrn~itv nail intar.nnna.+iv. mnr+ mfihila ThP Pffw+iTw .=..” ~r.ta ”.. U.W, - -“ “...JW.. . w y“. ””.., fluid-filled pores. The term “solid phase” as defined in this study includes both real solid grains and uncormective (isolated) pores. --1:2 ..-1 --:... --I_--l ..-1 --;... DA*LA..:A vG1ucxLy v~ dJG lUW.I D(JUI llLUU Vclw(my v and bullu
Wv-w.., . w U,llu ,Lm””.,w
‘(P;VP)=+P(;%+:% ---(8)
Expanding the first term of solid mass balance equation, Eq. 2, and applying Eq. 7 gives v.vs=– 1 (1–+)0. d[(l-+)ps] dt . ...............................(9)
volume averaged values with respect to a stationary coordinate frame. $V is equivalent to the fluid bulk volumetric flux (fluid-flow rate per unit bulk area). Similarly, (1-$)v, is the solid bulk volumetric flux. Note that for a deformable porous medium Darcy’s law is expressed as fluid
For constant solid mass and noting that +=VJVb and Vb=VP+V, where VP, F’s and Vb are pore, solid, and bulk volume, respectively, Eq. 9 is equivalent to v.v, 1 dVb =—— . ......................................................(lo) Vb dt
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SPE30752
H.Y. CHEN, L.W. TEUFEL, R.L. LEE
3
Thus, the divergence of solid velocity simply reflects the rate of change of bulk volume. Note that Eq. 10 is nothing more than a statement of solid mass balance, since it is derived exclusively from Eq. 2. From Eq. 10, v~=O(nondeformable) implies dVb=O. Applying the relation d@$=dV#VP-dV#Vb and Eq. 10 to Eq. 8 results in
v.
()[
~;vp
I dVP
=p($ L*+—— .
pdt
VPdt
. . . . . . . . . . . . . . . . ,....(11)
1
Eq. 11 is a key fundamental equation in this study. The right-hand-side of Eq. 11 basically represents the rate of change of fluid density and pore volume. As will be shown later, different interpretations of the pore volume change due to different boundary conditions result in different governing equations and the associated “total (or system)” compressibilities. A proper interpretation of the pore volume change, in fact, is critical in achieving fluid-flow and geomechanics coupling. Eq. 11 with d(.)/dt replaced by ~(.)/dt would be the result obtained from Eqs, 1 through 3 with v~=O (nondeformable). This nondefonnable case will also be discussed later. The change of fluid density term alp/p in Eq. 11 is related for fluid compressibilityy c by the following relations:
# =
at
pat’
1ap . ~vp=lvp;
P
CQ= Q3 ........(n)
dt P dt
chemically inert with the solid phase. (The assumption of constant solid mass has been invoked in deriving Eq. 10.) c~ would be the compressibility of pure solid grains without “dead” pores if the pores are perfectly connected and if the fluid is inert with the solid phase. Additionally, porosity is assumed to be constant during the unjacketed experiment which requires that the solid phase satisfies the condition of homogeneity. Obviously, a “solid phase” which includes dead pores can not satisfY the homogeneity criterion locally. Despite all these restrictions, the assumption of using unjacketed bulk compressibility to represent the “pore compressibility” shown by the second partial derivative in Eq. 13 is considered to be practically reasonable. It should be mentioned that the unjacketed bulk compressibility is a measurable quantity and the experiment is routine in practice. Through the reciprocal theorem of elasticity, Geertsma8 showed that the first partial derivative in Eq. 13 can be expressed in terms of the unjacketed bulk compressibility c~ (=1/KJ described previously, and the “drained” jacketed bulk compressibility cb (=l/Kb) measured under hydrostatic pressure (see Fig. 1 and Appendix A). A drained condition means constant pore-pressure during the test. A more detailed discussion of c,, cb, and other pertinent rock compressibilities are given in the Appendix A. In essence, the change of pore volume given by Eq. 13 may be expressed in terms of porosity and two measurable compressibilities, c~and Cb,as dV +$=— dVP _ - -(Cb -C,
Consider the change of pore volume term dV/V in Eq. 11. Following Geertsma,8 and Brown and Korringa, lfdVJVP can be expressed in terms of two partial derivatives as
P
Vb
)@d ‘I@@>
-c, )dcrm, ...........(14)
‘[cb ‘(l+$)c~]@+(cb
where Umis the mean (average) normal stress which is equal to the negative of confining pressure, i.e., am=–pC. (Also recall dpd =dpc–dp.) Introducing Eqs. 12 and 14 into Eq. 11 results in Here, pd is the differential pressure, PFPC–P, where p. md p are confining pressure and fluid pressure, respectively. In other words, an imposed confining pressure dpCis considered to be the sum of two incremental pressures, dpd and dp, i.e., dpC=dp&ip. The second partial derivative in Eq. 13 is assumed to be ..-.: --l. -.-a 1...11. -A--a ””n.:l:+.r ,. AL(—1 IV \ m.mmwd h,, LUG UUJitUIWICU UUIA UU1ll~l GSMUUILY b~ (— L~~~~) ,,,~-~wu UJ allowing the fluid to penetrate the connected pores such that the fluid pressure acts fully on the “solid phase” (see Fig. 1 and Appendix A). Under such a condition, the change of con!ining pressure is equal to the change of pore-fluid pressure, i.e., dpC=dp, and hence dp~O or p~const. Constit solid mass is assumed which implies that the fluid is dp ‘, dp 1 dt ..................................................................................(15) V. ~Vp P
()[
[)
[
= $c+cb–(l+@)c,
+(cb–c,)—
do.
or v. ‘k’k ;Vp +;~vp)z +(cb
= $c+cb -(l++)c,
‘C.)=
dc.
1
~,
dp
......(16)
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where the primary variable is fluid density p and fluid pressure p in Eqs. 15 and 16, respectively. Eq. 15 or 16 is a single expression derived from Eqs. 1 through 4 and 14. Constant fluid compressibility c is further assumed in Eq. 15. At this point, two major and reiated features in Eqs. 15 and 16 should be noted as compared to the conventional reservoir modeling (see, e.g., Ref. 18). These are: (i) the material derivatives, d(.)/dt, and (ii) the term dcr#dp. Item (i) evolved from the consideration of deformation (v@). Assumptions will be made soon to simpli~ the mathematical model. Item (ii) evolved fkom the interpretation of pore volume change (see Eq. 14). Interpretation of daJdp requires the consideration of the porous mechanical issues w~ch will be discussed later. We have found using fluid density as the primary variable results in certain discouraging features (mainly nonlinear terms) in a coupled mode. Therefore, we will simpli~ Eqs. 15 and 16 using fluid pressure as the primary variable. Eq. 15, however, is preferred under certain conditions which will be presented later. Simplified Pressure Equation. The following two assumptions are imposed for Eq. 15. (i) The fluid is slightly compressible such that p=pOexp[c(p-po)]zpO[ l+c@-po)]. This is also equivalent to linearizing Eq. 4 as C=(l/po)~ph3J. (ii) The material derivatives can be approximated by the partial derivative, i.e., d(.)/d=~(-)/&. This is equivalent to neglecting the dot product term in Eq. 7 by assuming v~.V(.)<@(-)/~t. Physical interpretation of this approximation is that the medium is undergoing deformation but remains -----phs stationary. For Eq. 16, ittni (ii) is also dSSLUIICd neglecting the quadratic term c(Vp)2 (which is also a dot product). With the above assumptions, both Eqs. 15 and 16 reduce to
consideration of the stress field which will be provided by poroelastic theory described next. Linear Poroelastic Theory pGi&n]=~jc ~~eory, utiou..ulu~ rlaoonh;”n
fi,.iA-cnli AEulu-.”’z”
A
m-mnlino .“.y.-.~
W%lQ r. -
developed in a series papers by Biot. 1-7 Perfect elastic medium (in the sense of linear, reversible, and non-retarded mechanical behavior) with small strains are assumed in this study. Isothermal condition is also assumed. Basic Relations. The three basic principles of poroelastic theory are: stress equilibrium, strain-displacement, and strain-stress-pressure relations. The y are parallel with the L-1 —--. . . nn . . . . ..$”1“.., . ...4 - . . . . ..+i-.. -C .+-+n# th IIItiSSUUJaJIGC, UdIby ~ law, ~u GqULIWU U1~=.u u. -e.1 +%;A. ~uflow modeling. Mathematically, these are: 1 Stress equilibrium : (6 equations)
3 *V z— =0;
j=] ‘j
‘ij =‘ji?
... ...... ..... ...... ..... ........ (19)
Strain-displacement relation : (6 equations) 1
&v. .—
2 &j
[)
i%li+hj
—
axi
-(20) . .........................................
Strain-stress-pressure : (6 equations) 1 . ~ii = -[aii E a [71a) ~v(cJj +gkk)]+—— n ...............\.. . 3Kb P ‘ (i #j). ..................................(21b)
SV = au /(2G),
v. ;Vp where
()
=+ct,l~+(c~
@
–c,)+
h
. ....................(17)
+ct,~= $C- (l+l$)c, +c~ . .......................................(18)
In Eqs. 19 through 21, SUand Oti are the components of bulk strain tensor and total stress tensor, respectively, ui is the component of solid displacement vector U(UX, UY,UZ),E, G (=E/[2(l+v)]), and v are Young’s modulus, shear modulus, and Poisson’s ratio for the solid skeleton under drained conditions, respectively, Kb (= l/cb) is the drained jacketed
L..ll. OUIK
muuluus
—-
A..1...-
as u~lmcu
--
A
C-...a
idit3i,
iiLIU
.. ..A-
W
:. +La
1S
LUG
-A-fial.”+;,.
~ULUGICWLIU
Ct is a commonly used notation to denote “total (or system) compressibility.” The added subscript I is used to distinguish different c~s presented in this paper. For isotropic and homogeneous permeability (kconst.) and constant fluid viscosity w, the left-hand-side term in Eq. 17 reduces to V.[(klW)Vp]=(Hp) V2p where @ is the Laplacian operator (V%#/x2+&’$+#/z2). A relationship between p and IS. is required to define Eq. 17 (or 15, 16) completely. This relation requires the
parameter or effective stress coefficient. A more detailed discussion of c.twill be given later. Body forces and inertial effects me neglected in Eq. 19. Small strains are implied in Eq. 20. The stress and strain are taken positive in tension .-—. ---:_whereas fluid pressure p is positive for compreswm. Note that fluid pressure p affects normal strain only and in the same manner due to the assumption of isotropy (see Eq. 21a). Shear strains area fimction of shear stresses and are independent of fluid pressure (Eq. 2 lb).
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H.Y. CHEN, L.W. TEUFEL, R.L. LEE
5
It is more convenient to express stress in terms of strain because the total stress satisfies the equilibrium equation @q. 19). Solving Eq. 21 for stress givesc oti =2GkV+le6V –ap~u, .......................................(22)
Governing Equation. The total stresses given by Eq. 22 must satisfi the equilibrium relation specified by Eq. 19. Introducing Eq. 22 into Eq. 19 and applying straindisplacement relation (Eq. 20) give
where 3U is Kronecker’s delta (8ti=l for i=j, 13J=0for i#j,), and ~ is the Lame’s constant which is related to other mechanical properties by 3vKb k=—=—= I+v 2vG 1-2V Kb –:G. .................................(23)
..(29)
These are three equations in x, y, and z. Simplifying Eq. 29 for constant elastic coefficients, G, h, and a gives GV2ui+(G+A)~=a hi 13p —. ...................................(30) ~i
Adding the three equations given by Eq. 21a or 22 results in e= where e =&u +&H +S==, ..................................................(25) am = (on +6W +Ozz) /3. .....................................(26) Here, e is the dilatation or volume strain of solid skeleton and am is the mean nonnai totai stress. Eq, 24 in fact is closely related to Eq. 15 where the interpretation of rock compressibilities were presented (see also the discussion pertinent to Eq. A-9, Appendix A). A generalized form of Eq. 24 covering three types of boundary conditions, uniaxial, biaxial, and triaxial strain, is presented by Eq. B-2 of Appendix B. The solid displacement velocity v, and the volume strain e are related to solid displacement vector u by v~= war; e=v.~ . .............................................(27) am +Ctp L+(2/3)G = Cm +ap Kb ) .....................................(24)
Note that G+L=G/(1-2v). Eliminating the mean stress am between Eqs. 24 and 17 results in V. ~Vp P where
()
=$ct,ll~+a~,
..................................(31)
@t,II = 4W:I- a ‘cb = $c +(a -$)c z? .......................(32) and a=l –c~cb. $ct,ll is equivalent to Biot’s modulus M.4’7 Eqs. 30 and 31 are four equations in four unknowns, p, Ux, Uy, and UZ.(Note that e=V.u, see Eq. 27.) This coupled system governs the time history of the deformation and the pressure field. By adding the three equations of Eq. 30, a compact representation is GV2u+(G+k)VV. u=aVp , ..................................(33)
The second expression, e=V.u, may be established from Eq. 20. From Eq. 27, it follows that the divergence of solid velocity V.v~ (Eq. 10) is related to the volume strain e by Vv s =~_ dt ldVb. Vbdt’ de=-. ........................(28)
vb
Eqs. 31 and 33 forms a system with basic variables of scalar p and vector u. (Again note that e=V.u.) Alternatively, we may select p and e as the basic variables. Taking the divergence of Eq. 33, which is equivalent to differentiating the three equations of Eq. 30 with respect to the corresponding xi and then adding the resulting three equations, results in (1+ 2G)V2e = aV2p. ..............................................(34)
I
The first expression in Eq. 28, again, is a statement of mass conservation. The second expression, which follows directly from the first expression, provides a straightforward interpretation of the volume strain e. Eq. 21 or 22 provide a timctional form between strain, stress, and fluid pressure which is required for solving Eq. 15, 16 or 17.
V2e=0 if a=O which corresponds to a nonporous medium. Eqs. 31 and 34 are two equations in two unknowns, p and e. The term I+2G in Eq. 34 basically is a modulus related to other elastic constants by
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1 — I+2G
1–2V _ 1 —.. “n l-v
cb l+v — 3 l–v
= Cb’, .......................(35)
where Cb’ is used to simpli~ the presentation. Note that cb’+cb ~ v~().5. cb’ is the compressibility co~espon~g to an uniaxial strain condition without fluid pressure effect (e.g., GH=sB=O, EZZ*O, andp=O; see Eq. B-2, Appendix B). In summary, we have shown, starting from the basic principles, how the fluid-flow and geomechanics are coupled. Two sets of coupled equations are presented: (i) Eqs. 31 and 33 with variables p and u, and (ii) Eqs. 31 and 34 with variables p and e. Discussions The forgoing derivations indicate two major and related concepts in achieving the coupling between the fluid-flow and geomechanics. The first is the interpretations of various rock compressibilities as discussed in Eqs. 13 and 14, and Appendix A. The second is the fimdamentd stress-strainpressure relations shown by Eqs. 21, 22 and 24. Implicit in these relations is the concept of effective stress introduced by Terzaghi19 in 1923. The ‘presented treatment of the rock which essentially follows that of compressibilities, Geertsma,g certainly is not the most detailed description. At present time, however, we feel the essentials are included in our considerations. In fact, a further detailed modeling of rock compressibilities will not alter the structure of the governing equations. Two coupled pressure equation were presented, Eqs. 17 and 31. Both reduce to a decoupled difision-type equation if crm(mean normal stress) and e (volume strain) are constant with respect to time in Eqs. 17 and 31, respectively, assuming ch#c~ and cd. (T’he term “decoupled” implies a closed equation with a single dependent variable.) The compressibility terms involved, however, are different. If am ore are constant in time, the decoupled pressure equation can be solved independently of the stress field. This is the most favorable case tlom problem solving point of view. In the following, the concept of effective stress is discussed first. Two special cases are then discussed: se--, ,. nonaerormarne and defomialie witt consUT mean normai stress. Finally, some additional features of the pressure equations are examined. Effective Stress Concept. The ej]ective stress, denoted as csJ, may be defined as20 0; =(JU +api5u. ......................................................(36) This effective stress represents the portion of the total stress which is in excess of some fraction of the stress caused by
fluid pressure. The original Terzaghi’s effective stress is defined as aq+pbti, which is equivalent to ct=l in Eq. 36. As will be shown immediately, the definition given by Eq. 36 essentially eliminate the explicit role of fluid pressure in the constitutive strain-stress relations. This, in turn, offers a simple interpretation of Biot’s poroelastic theory. It should be pointed out that Biot1}2did not explicitly use the concept of effective stress to develop his theory, now commonly called the theory of poroelasticity. Eq. 21 in terms of the effective stress (Eq. 36) is
Eii
..................................(37a) =[a~ ‘v(~~ +O_~k)l/~,
Sti =a;
I(2G),
(i %)], ..........................................(37b)
which essentially is the Hooke’s law for an elastic nonporous body. 21 Similarly, Eqs. 22 and 24 can be expressed as 0; =2G%V+Aea~, ...................................................(38) and
e ~=~m/&b,
Iv
(9 ll\ .............................................................(37)
e is the effective mean normal stress respective y, where crm defined as CT; =Um +Ctp =(C2 +6;Y +o~) / 3. ....................(40) Both Eqs. 38 and 39 are the expressions in the classical nonporous elasticity. 27 30 Aiwlnw @Y~e rna~~? ~rn.nnrtmt l%c =.. —. -y”. z , thrnnoh -----. . . . . . . . . t~~ feature in the theory of poroelasticity, namely, the concept of effective stress. Specificallyy, the constitutive strain-stresspressure relations are assumed to follow the classical nonporous elasticity if the stress are replaced by an “effective” stress which, in some kind fashion, incorporates the effects of fluid pressure. Implied in the above interpretation are two major issues. First, the eiastic behavior and the mechanical properties of a porous fluid-bearing rock are assumed to be governed by the effective stresses alone (a single vtiable), instead of total stress and pore pressure (two variables). Second, what is the functional form of a? In other words, how to “weigh” the effect of pore-pressure. The first issue will not be discussed
h~r~ -w. -.
in
.A. Aiwlnminn -“” ------
nf .,.
the --
~emmi-1 “., V.Z-..
imm= . -----
hnwever --- . . - . -.,
ic --. nPrPmmv -””.-., -“
the context of this study. Essentially, a determines the relative contribution of pore-fluid pressure on the elastic bulk behavior of a porous medium. It is not difficult to image that attempting to define an exact and unique a is an extremely difficult task, given
512
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H.Y. CHEN, L.W. TEUFEL, R.L. LEE
7
the inherently complex nature of a natural porous fluidhe~r@ r~~k (e.g., pore ~~~~~~re, r~~,k ~~n~ituent~ .- —-.. anisotropy, nonhomogeneity, multiple fluid phases, etc.). Nevertheless, it is instructive to know the deterministic nature of ct and the associated limiting values under rather idealized conditions. The exact expression of a, from the bulk volume strain point of view, is7’8’20
(1
ASpointed out ear!ier Eq. 42 also can be derived from Eqs. 1 through 3 with v~=O and d Vb=O. For constant c (fluid compressibility), Eq. 42 maybe written as V. fv’p where
()
=$(c+cpb)~
, .....................................(43)
=l-(~~/~~)=l–(c~/cb),
.................................(41)
where Kb (=l/cb) and KS (=1/cJ are, as defined before, drained jacketed and unjacketed bulk modulus, respectively. Eq. 41 implies that a will not be constant if KJK, is not constant. In the theory of linear, isotropic poroelasticit y, a is considered a constant. Therefore, the effective stress given by Eq. 36 is a linear combination of total stress and fluid pressure. Eq. 41 indicates that a is dimensiordess and is bounded by O<ct<l (by taking c.#-0, c@ and cb>c~). Biot and Williss fiuther argued that the value of a can not be smaller than porosity, i.e., $<cc<l. ct+l as c~cb+() such as the case of incompressible solids (c~+.0). ct+O as c~cb+.i (e.g., $+ti) and the effect of pore pressure (h last term of Eq. 36) vanishes. It should be pointed out that different physical interpretation of cx are required for different physical Eq. 41 is derived fkom a bulk quantities and processes. *0$*223 volume point of view (see also Eq, A-9, Appendix A). The exact expression of ‘7x” from q pore volume point of view is given by the (3 of Eq. A-13; Appendix A. Furthermore, uncertainties are present even when the same physical quantity is concerned. Warpinski and Teufe124’25noted poor agreement between experimental and theoretical a values. They also noted a plausible result in Ref. 25: a trend of decreasing a with decreasing permeability and porosity of carbonate rocks (chalks and limestones). At present, it appears that a is best treated as a material property similar to permeability and porosity. This implies that an experimentally determined ct value under the proper scale should always be honored. Nondeformable Porous Media. The condition for a nondefonnable medium is v~=O,i.e., zero solid velocity. The consequences of v~=Oare: (i) d(.)/dr=~(.)/~t (see Eq. 7), (ii) dVb=Oand de=O @q. 28), and (iii) dcrm=-adp (Eq. 24 or AT&w item (i) Rn 11 rqjnces tn 0) I .). w.~a ---\./, -y. . . .---., .-
cpb=j+$)v,=; (:)6, ...........................
..(44.
and c is defined by Eq. 4. cPb (Eq. 44) represents the pore compressibility under constant bulk volume (d Vb=()). Eq. 43 indicates the fluid density satisfies a decoupled diffusion-type equation for a medium of constant bulk volume but with “pressure-dependent” porosity. (No restrictions were imposed on the permeability in Eq. 43.) The fluid density p in Eq. 43 can be replaced by fluid pressure p if the assumption of slightly-compressible-fluid is further imposed (see the assumptions used to obtain Eq. 17).
T“ H? ‘servoL* . elgxNxlTJl . $3, ~ $(jt~ (or ~.,.
w.dem) “= “---..,
compressibility, commonly denoted as c!, is used to represent the total contribution from fluid and rock compressibilities. The boundary conditions of the rock compressibility, however, is generally not explicitly specified. For example, Ct=c+cfis defined in Ref. 18 where c is given by Eq. 4 and cF(v~)*lap In the context of Eq. 43, the total compressibility would be c@+@ T’bus, one possible interpretation of the rock compressibility commonly used in reservoir engineering is the cPb defined by Eq. 44 where the boundary condition is a constant bulk volume. Consider Eq. 15. Imposing dam=-adp and d(.)/dr=a(.)/at (for v,=O) to Eq. 15 results in
v. :Vp
()
=l$ct,ll ---,
ap
............................................(45)
where I$ct,ll=$c+(ct-$)c~ (see Eq. 32). Comparing Eqs. 43 and 45 suggests the following relation between Cpband c~,
Cpb =
[(~ – 1$)/+]cs . .................................................(46)
+;+=P’JL2+;9
(pa p-’)
. .,....................(42)
Same result can be obtained from Eq. 14 using dam=–m@. :------ A Note tit kp#[(i/w)-~]c$ beta-use $Su<l as hwusseu earlier. Eqs. 43 and 45 are the “exact” fluid-density equations for a nondeformable porous medium.
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COUPLED FLUID FLOW AND GEOMECHANICS IN RESERVOIR STUDY
SPE 30752
llafnrmnmhlcm US.”* U-”,W
Mmlim . ..-”.-
Wi+h . . ..U
rnnatnnt -“------
Mman ..1*”..
Nnrmal . .“- .
. .
Qtroac “.. w“”.
The conditions are v,#O (and e#O) and constaut am (i.e., dam=O). Applying dam=O to Eq. 15 and assuming d(-)/d@”)/ii result in
v.
[)
P
(k \ —Vp =($C*,lll:+aci~
3—
ar
, ..............................(50)
‘@p,
()
$ct,J~~ =+ct,J~ +cX2cj = $C+(cx -+)C, +azc~ . ........(51) =$cf,! ; , .............................................(47) $Ct,lllis equivalent to Biot’s modulus h4c.6 As can be seen from Eq. 50, the fluid pressure will not obey the diffusion equation unless f is independent of time, i.e., a@t=O. If the condition of aflt3t=0 is satisfied, the fluid pressure then follows a diffusion-type equation with a porosity-compressibility term given by I$ct,lll(Eq. 51). Such a decoupled diffusion equation then can be solved independently of the stress field. The fiction J which evolves from the integration of Eq. 34, will depend on the boundary conditions of e and p. Fourth-Order Pressure Equation. Assume constant values of W.Land $Ct,ll in Eq. 31. Apply @ operator to Eq. 31 and -1; —:-,-.. t77.. -:-129,4 m.. --...1. :AL -—~lmlllldl~ men v -e using J2q. J+. L Ilc IGSUI1 1s b III av2P V2V2P . __ at klp
where I$ct,l=$c+c~-(l +L$)cs(see Eq. 18). From Eqs. 11 and 14, the rock compressibility contained in $ct,l represents 1 avp c~ –(l+$)c~ —— = Vp ap . . $
()
= Cpc Y .....................(48)
where CPCis used to denote the pore compressibilityy under Constant COIIfh@ pressure (dpC=–dom=O). ThUS, c~,l dSO may ‘be expressed as ct,l=c+cpC.CPC defined by Eq. 48 oilers another possible interpretation of the rock compressibility commonly -—=—--.—= .-. .—.-— uSed in resemoti emzineeritw. Similar to Eq. 45, the fluid density of a deformable medium under constant mean normal stress also satisfies a decoupled diffusion-type equation as indicated by Eq. 47. The rock compressibilityy terms involved, however, are different due to different boundary conditions. Note that de=ctcbdp follows from Eq. 24 if dom=O. That is, the change of volume strain is proportional to the change of fluid pressure (with constant values of a and cJ. Muskat26 showed the fluid density in a nondeformable porous medium satisfies a homogeneous diffhsion equation if permeability, porosity, vis~ty, and fluid compressibility are constant. (Ms result is equivalent to Eq. 43 with constant ti~ and cPb=O.) Eqs. 45 and 47 formally extend the applicability of Muskat’s fluid-density formulation by includine a rock _–––= --– compressibility defined by Eqs. 44 and 48, respectively. Eqs. 45 and 47 are derived under the assumptions of non-deformaiie medium and constant mean normal stress, respectively. We now examine the pressure equation and the associated total compressibility for a deformable medium without restriction on the stresses. Pressure Diffusion Equation Under General Conditions. Following Verruijt,10 integrating Eq. 34 results in e=c~[ap+ f(x,y, z,r)], ...........................................(49)
‘ ““””” ””””””.”””’”””””””””””..”
(52)
where $C1,lllis given by Eq. 51. Eq. 52 also can be obtained from Eq. 50 by taking V2 operator and noting that V2fi0. Eq. 52 is a fourth-order equation in terms of p and a second-order equation in terms of V2p. Except for the notations, Eq. 52 agrees with Biot’s3 result derived from the method of stress fimctions. It should be emphasized that constant value or small change of porosity is implied in deriving Eq. 52. The exception is when $Ct,ll=Ocorresponding to incompressible fluid and solid (see next section). Our intention of presenting tha nntmna nf tha m-~ccnrcnlntifin fnr n K“ <9 i. tn hinhlinht Uq. ak no .“ ul&llA&l. .=. ---“. u. y. . ..-. .“ . ...-”.. .. . deformable medium. From Eq. 52, the fluid pressure is the sum of two functions, p=p1+p2 where PI and P2 have to satisfy, respective y, 3,4 v2p1 = *~ klp and V2p2 =0. ...................(53)
at
where Cb’is given by Eq. 35 and J has to satisfi LaPlace’s equation V2fi0 for every value of q i.e., yfmust be harmonic. Applying Eq. 49 to Eq. 31 with constant k/w gives
Thus the commonly used diffhsion equation is only a subset of the entire true set. Same implication also can be inferred from Eq. 50. The fluid pressure, however, follows a Whether homogeneous diffusion equation if p2=0 (i.e., p=pl). p2 vanishes or not will primarily depend on the boundary conditions. Note that any one of the following variables, fluid pressure p, volume strain e, and mean normal stress crm,may
514
SPE30752
H.Y. CHEN, L.W. TEUFEL, R.L. LEE
9
1--
. ..__..
--a
:_
Ue tmpIGsstu
m
.. - ,.4.- qL.-. Umnti U1 LUG
.A1. e.. ULUG1
+..,A Lwu
..,.fi+. aQbul
;=” dul~
ta
q.
24.
Therefore, relations similar to Eq. 34 also can be established for pairs of @e and V2am, and V2p and V%m. In other words, @p, @e, and V%m are related to each other linearly. Thus, Eq. 52 maybe expressed as
57. Whh!ecx?nmmt Inean !mmai stressis assutt.ed, Eq: 57 presents a simple model to consider variable permeability. Note that both Eqs. 56 and 57 do not involve porosity. Also, the compressibility terms in Eqs. 56 and 57 are different.
Conclusions The major contribution of this paper is showing how Biot’s isothermal, linear poroelastic, coupled two-phase (fluid and rock) modei is evoived from the conventional porous fluidflow modeling. Our primary goal is to preserve the Specifically, all p, e, and a= satis~ the same fourth order commonly used systematic fluid-flow modeling so that future equation with the same difl%sivity. This implies that p, e, and modeling of complex reservoir situations involving crmmay share the same fundamental solution. They may be geomechanical issues (e.g., naturally fractured reservoirs, stress-sensitive reservoirs) can be formulated more different only through the initial-boundary conditions. systematically and easily. The presented approach offers an to non-geomechanical alternative avenue (especially Incompressible Fluid and Solid. For incompressible fluid engineers) to recognize the relationship between the fluidand solid (c=O, c~=O),we obtain ~ct,l=c~ (see Eq. 18), $Ct,ll=O flow in a nondeformable field and Biot’s coupled flow(see Eq. 32), and 0Ct,111+2C~(see Eq. 51). Note that m=l if deformation field equations. In addition to the coupled field its explicit form given by Eq. 41 is used. Above limiting equations, some decoupled equations are identified under conditions may be applied to appropriate equations presented various assumptions. These decoupled equations have before. Two decoupled equations are discussed next. potential to be solved independently of the stress field. Eq. 31 with I$ct,ll=Oand constant I@ reduces to Major conclusions drawn from this study are: (1) Consistent interpretation of rock compressibilities and the effective stress law are the two most important steps v. ~vp zag. ..................................................(55) P required to couple fluid-flow and deformation fields. (2) At the fimdamental level the fluid-flow governing equation is the same for both deformable and nondeformable It can be shown that Eq. 55 is exact (within our assumptions) media, except that the former requires a total derivative if deli% is replaced by deldt. Eq. 55 reduces to the Laplace interpretation of the rate of change of fluid density and pore equation ~p=O if de=O (nondeformable) and constant k@. volume (see Eq. 11), whereas the latter only requires a partial Applying Eq. 34 to Eq. 55 with constant ld~ results inl’2 derivative (see Eq. 42). (3) Whether the fluid density (or pressure) satisfies a a2cj & decoupled diffusion equation or not depends on the boundary V2e = —— (56) klp 13 ‘ """"" """"""""""""""""""""""""""""""'"""""-"-""""`"""" conditions of the problem at hand (see Eqs. 45,47, and 50). (4) The fluid density satisfies a decoupled diffusion equation under two conditions: (ij nondefoiinabk niedi-m~l, where c~’ is given by Eq. 35. Note that azc~’S$c~,J1lif c=O and (ii) constant mean normal stress (see Eqs. 45 and 47). and c~=O.By Eq. 39, Eq. 56 is also satisfied by the effective (5) The “rock compressibility” involved in a given field mean stress crme.Thus, the bulk dilatation e and the effective equation also depends on the boundary conditions of the th~ hnmno~npnnc diffil~inn rw~~~~~ if ~,e= ~~e~~~me nhnv ““w, .ss” Al”... ”&v..-”-. . . . . . . . . . -q problem. This is important for a consistent interpretation both solid and fluid phases are incompressible. The limitation between laboratory, field, and simulation results. of Eq. 56 is that permeability and viscosity must be constant. (6) Under the simplest case (isotropic homogeneous However, no assumption is made concerning the stress state. material properties), the fluid pressure satisfies a fourth-order If we further assume the mean normal stress is constant, equation (see Eq. 52) instead of the conventional secondi.e., dan=O, in addition to c=O and c~=O,Eq. 16 reduces to order diffusion-type equation. The solution of the former includes that of the latter as a subset (see Eq. 53). (7) The volume strain satisfies a homogeneous diftision V. +Vp ‘cb~ , ..................................................(57) equation if the fluid and solid phases are incompressible and the medium is isotropic homogeneous (see Eq. 56). (8) The fluid pressure satisfies a decoupled diffision in which dp/dz4p/?lt is also assumed. Eq. 57 is a limiting equation if the fluid is incompressible and the mean normal form of Eq. 47 which is valid for compressible fluid and stress is constant (see Eq. 57). solid. In fact, c~=Ocan be relaxed to obtain the form of Eq.
()
()
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Nnmnndati . .W . . .”..”.-.”
Irn .w
~
I
c= e= E= f. G= k= K= M= P= t= u= v= v= V.>T= ~3Y9a= p= 6V ;: =
compressibility, Lt2/m volume strain, dimensionless Young’s modulus, rn/Lt2 function defined by Eq. 49 shear modulus, m/Lt2 permeability, L2, md modulus, m/Lt2 modulus, m/Lt2 pressure m/Lt2 time, t displacement, m/Lt2 velocity, L/t volume, L3 &~t~~~, ~ poroelastic constant @q. 41), dimensionless constant (Eq. A-13), dimensionless Kronecker’s delta (8.=1 for i=j, 8ti=0 for i+) strain, dimensionless porosity, fraction Lame’s constant, m/Lt2 viscosity, m/Lt Poisson’s ratio, dimensionless density, m/L3 stress, m/Lt2
k=
P= v p= 0= =
Subscripts ~=
h..lb U1.un
confining ;: differential “ m= mean pore P= pb = pore volume, with constant bulk volume pc = pore volume, with constant confining pressure s= solid total, system t= o= reference Superscripts e= f = It = !11 =
effective uniaxial strain ~~ax~~ ~tr~ triaxial strain
Acknowledgments The authors thank Phillips Petroleum, Union Pacific Resources, Vastar, Conoco, and Amoco for their support of this study. Discussions with H. Ruistuen were most helpfid during the writing of this paper.
1. Biot, M.A.: “General Theory of Three-Dimensional Consolidation,” J. Appl. Phys. (1941) 12, 155-164. 2. Biot, M.A.: “Theory of Elasticity and Consolidation for a Porous Anisotropic Solid:’ J. Appl. F%ys. (1955) 26, 182-185. 3. Biot, M.A.: “General Solutions of the Equations of Elasticity and Consolidation for a Porous Material,” J. Appl. Mech. (1956) 27,91-96. “Thermoelasticity and Irreversible 4. Biot, M.A.: Thermodynamics;’ J. Appl. Phys. (1956) 27,240-253. 5. Biot, M.A. and Willis, D.G.: “The Elastic Coefficients of the Theory of Consolidation;’ J. Appl. Mech. (1957) 24, 594-601. 6. Biot, M.A.: “Mechanics of Deformation and Acoustic Propagation in Porous Medial’ J. Appl. Phys. (1962) 33, 1482-1498. 7. Biot, M.A.: “Nonlinear and Semilinear Rheology of Porous Solids;’ J. Geophys. Res. (1973) 78,4924-4937. 8. Geertsma, J.: “The Effect of Fluid Pressure Decline on Volumetric Changes of Porous Rocks;’ Trans. AWE (1957) 210,331-340. 9. Jaeger, J.C. and Cook, N.G.W.: Fundamentals of Rock Mechanics, 3rd Ed., Chapman and Hall, London (1979). 10. Verruijt, A.: “Elastic Storage in Aquifers,” Flow Through Porous Media, R.J.M. De Wiest (cd.), Academic, San Diego, California (1969) 331-376. 11. Raghavan, R. and Miller, F.G.: “Mathematical halysis of Sand Compaction,” Compaction of Coarse-Grained Sediments, 1, Chap. 8, G. V. Chilingarian and K. H. Wolf (eds.), Elsevier Scientific Publishing Co., Amsterdam (1975) 403-524. 12. Rice, J.R. and Cleary, M.P.: “Some Basic Stress Diffhsion Solutions for Fluid-Saturated Elastic Porous Media With Compressible Constituents;’ Rev. Geophys. and Space Phys. (1976) 14, 227-241. 13. Rudnicki, J.W.: “Effect of Pore Fluid Diffusion on Deformation and Failure of Rock,” Mechanics of Geomaterials, Chap. 15, Z. Bazant (cd.), John Wiley 8Z Sons (1985) 315-347. 14. Detournay, E. and Cheng, A.H-D.: “Fundamentals of Poroelasticity~’ Comprehensive Rock Engineering: Principles, Practice and Projects, Vol. 2, Chap. 5, J. A. Hudson (cd.), Pergamon Press, Oxford (1993) 113-171. 15. Bear, J. and Bachmat, Y.: Introduction to Modeling of Transport Phenomena in Porous M-edia, Kiuwer Academic Publishers, Boston (1990). 16, Cooper, H.J.Jr.: “The Equation of Groundwater Flow in Fixed and Deforming Coordinates;’ J. Geophys. Res. (1966) 71,4785-4790. 17. Brown, R.J.S. and Korringa, J.: “On the Dependence of the Elastic Properties of a Porous Rock on the Compressibility of the Pore Fluid;’ Geophysics (1975) 40,608-616.
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.
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H.Y. CHEN, L.W. TEUFEL, R.L. LEE
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18. Matthews, C.S. and Russell, D.G.: Pressure Buildup and Flow Tests in Wells, Monograph Series, Society of Petroleum Engineers, Dallas (1967) 1,4-7. i9. Terzaghi, K.: Theoretical Soil Mechanics, Wiley, New York (1943). 20. Nur, A. and Byerlee, J.D.: “An Exact Effective Stress Law for Elastic Deformation of Rock With Fluids,” J. Geophy.s. Res. (1971) 76,6414-6419. 21. Timoshenko, S.P. and Goodier, J.N.: I%eory of Elasticity, 3rd Ed., McGraw-Hill Kogakusha, Ltd., Tokyo (1970). 22. Robin, P.-Y. F.: “Note on Effective Pressure,” J. Geophys. Res. (1973) 78,2434-2437. Stress for Transport 23. Berryman, J.G.: “Effective Properties of Inhomogeneous Porous Rock,” J. Geophys. Res. (1992) 97, 17409-17424. 24. Warpinski, N.R. and Teufel, L.W.: “Determination of the Effective Stress Law for Permeability and Deformation in Low-Permeability Rocks;’ SPE Formation Evaluation (June,1992) 123-131. 25. Warpinski, N.R. and Teufel, L.W.: “Laboratory Measurements of the Effective-Stress Law for Carbonate Rocks Under Deformation,” ht. J. Rock Mech. Min. Sci. & Geomech. Abstr. (1993) 30, 1169-1172. 26. Muskat, M.: The Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill, New York (1937) 121-136. Appendix A - Solid, Bulk and Pore Compressibilities This appendix describes the physical interpretation of the unjacketed bulk compressibilityy c~ (=1/K~) and the drained jacketed bulk compressibility Cb (=l/Kb). Bulk, pore, and solid volume changes are describeti. The connection of c~ and Cb to solid mass balance equation @q. 10 or 28), Biot’s ccmstitutive strain-stress relation @q. 21), and the effective stress concept are also discussed. Two well-known basic relations relating various volume changes of a porous medium are dVb —=47 Vb and dvp dV$ +(1–+)— >.....................................(A-1)
jacketed tests. The experimental conditions conditions) of these two tests are described next. Unjacketed Bulk Compressibility. compressibility is defined as
(boundary
The unjacketed bulk
1 dvb . ......................................... ,....3)-3) c~=– — — ‘b 8P Pd Here, as shown by Eq, A-3, the experimental condition is a constant differential pressure pd which is p~pc–p where PC and p are the confining pressure and fluid pressure, respectively. The measurements are the changes of bulk volume asp changes. Note that the condition of constant pd means dp~O or dpC=dp, i.e., the change of confining pressure is equal to the changeof fluid pressure. c~reflects the compressibility of the solid phase. Geertsma,8 Biot and Willis,5 Brown and Korringa,17 among others, argued that the porosity remains constant during an unjacketed test only if the solid phase is homogeneous. From d$=O (constant ~) and Eqs. A-1 and A-2, the following relation can be derived: dvb —= Vb dvp _ dV~ — -—. . Vp V,’ (b= const.). ........................(A-4) . . “ ‘
[1
With Eq. A-4, Eq. A-3 also can be written as
(I$= const.). .....................................................(A-5) Eq. A-5 indicates that c. is a solid, or pore, or bulk compressibility defined under unjacketed condition if the porosity is indeed constant. (cS is equivalent to Geertsma’s cr.) Eq. A-5 allows the change of solid or pore volume to be inferred by measuring the change of bulk volume. Drained Jacketed Bulk Compressibility. The drained jacketed bulk compressibility is defined as8’17
P
v,
d$ dvp dt’b —= —.— , ...................................................(A-2)
where Vb, V,, and V are the bulk, solid, and pore volume, respectively, $ is ~e porosity, $=V~Vb, and Vb=V,+Vp. Geertsma8 showed that the interpretations of dVb, dVP, dV~, and d$ in Eqs. A-1 and A-2 may be achieved dlrougil two basic compressibility experiments: unjacketed and drained
? ..............‘A-6) “=-*[3%-[21P
where fluid pressure is maintained constant (i.e., dp=O) during the test and the confining pressure is hydrostatic. From dp=O, it follows that dp~dpC. The measurements are the changes of bulk volume and cb reflects the compressibility of the rock frame structure.
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COUPLED FLUID FLOW AND GEOMECHANICS IN RESERVOIR STUDY –($c, )@+acbdam,
SPE 30752 ............(A-12b)
Bulk Volume Change. Following the approach given by Refs. 8 and 17, an incremental confining pressure dpC is The change of bulk expressed as dpC=dp#p where p~pc–p. volume d Vbdue to an imposed @C thus can be considered as the sum of two incremental pressures, @d and dp, i.e.,
=
(acb
where Om=–pc. Eq. A-12 also can be expressed as d J(P_ Cicb —(dam ~$ +13dp); ~ = l-~. ~bt$ ..........(A-13)
ADDIVinQ c- fEo.- -––, A-3) and c. (Ea. A-6) to Eq. A-7 and noting ––rr–, —a–s,—z ——–”,—-L -– -, results in that dp@pc-dp dVb ‘—= v~ cbdp~
Eq. A-13 is an effective stress law similar to Eq. A-9 but lmtthe nhvsical noint fi~rn.fh~ pore VQhWrn.e =.—. of -- Vkw, 23NOte -. _.-. t —-.-—. ~—,-.-— interpretation of ac~$ is given by Eq. A-11. Solid Volume and Porosity Changes. Given dVb and dVP (Eqs. A-8, A-9, A-12 and A-13), dV’ and d$’can be obtained easily from Eqs. A-1 and A-2, respectively. Appendix B - Uniaxial, Biaxial, and Triaxial Strain Generalized equations for uniaxia.1, biaxial (plane), and triaxial strain are presented in this Appendix. General Equations. Assume that the strains (and displacements) occur in one axis only (say, z-axis, uniaxial strain), or in two axes (say, y- and z-axis, biaxial or pwe strain), or in all three axes (triaxial strain). These can be described as9 Uniaxial: Biaxial: Triaxial: en= SYY = O,s= = e # O, ......................(B-la) Cu = O, eYY+cZZ=e*O , .....................(B-lb) &H+ EYY + &== e # O> ..........................(13-lC)
+C#@=Cb(dpc ‘~dp), ..................(A-8)
where ~l–c~cb (see Eq. 41). Eq. A-8 relates c~and cb to the change of bulk volume. From Eqs. A-8 and 28 (de=dv~vb), it follows that dvb de= —=–cb(@C vb –czdp) =cb(d~m +w@), .......(A-9)
Eq. A-9 is the differential form of Eq. 24, where Gm=–pc. =(~mtiP)/K@ which evolved !kOm Biot’s COnStitUtlve strain-stress relation @q. 21a). One can easily veri~ that Eq. A-9 reduces to Eq. A-6 when dp=O (or p=const.), and reduces to Eq. A-3 when dpc=dp (or dp~O, p~const.). The derivations of Eqs. A-8 and A-9 provide the functional form of effective stress law ffom Z,he bulk volume point of view.23 Pore Volume Change. Similar to Eq. A-7, the following expression can be written ford V#/VP,
in which the definition of e=Eu+sYY+szZ is always followed. The case of uniaxial strain is a commonly adopted assumption, e.g., no lateral movements due to horizontal crustzd constraint. Biaxial (plane) strain is frequently invoked in studying stresses around a deep wellbore. Triaxial strain
deriva~ive can b; expressed in terms of c~ “@q. A-3) &d Ch @q. A-6) asg
simultaneously to highlight some systematic features caused by the imposed strain dimension. From Eqs. B-1 and 21, it can shown that
where a= i ‘(c~cb,). Therefore, Eqs. A-1 Omay be written as dVp _ —-++’ 17
‘b
.—
---
Uniaxial: o~ =On; 1 c;=—= 1.+2G l+V ‘cb, 3(1-v) ............(B-3a)
dV
~cbdpd
‘$c@,
.......................(A-12a)
‘P
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H.Y. CHEN, L.W. TEUFEL, R.L. LEE
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Biaxial: 0; = aYY‘o= 2’ Triaxial: 0: = 6 22+ayy +crzz 3 ;C; = 1 A+(2/3)G = cb . (j3-3c) Thus, all three compressibilities approach the same value as the Poisson’s ratio v approaches the upper bound of 0.5. S1 Metric Conversion Factors Cp x 1,0* E–03 = pas E–Ol=m II X 3.048” ft2 x 9.290304” E-02= m: ft3 x 2.831685 E–02= m E+OO=cm in x 2.54* md x 9.869233 E–04= pm2 psi x 6.894757 E+OO=kpa
*Conversion factor is exact.
.
c; = fi
= ;(l+v)c~,
...(B-3b)
From Eq. B-5, the following bounding and limiting cases can be obtained:
Here, the superscripts, ‘, “, and ‘“, represent uniaxial, biaxial, and triaxial strain condition, respectively. o ~ (n= ‘, “, ‘“) represents a mean total stress where averaging is taken only for the stress with which the strain is not vanishing. Eq. B-2 for the triaxial case is Eq. 23. Note that a # =am and c~=cb. In terms of effective stress form, Eq. B-2 becomes
n en. ‘=cbom ~ ‘; ‘a; ‘sp; (n= ‘, “, ‘“).
.........(B-4)
The three compressibilities in Eqs. B-3 are related by
c;=—
1 2(1-V)
c;
= —c;.
I+v
3(1-V)
.................................(B-5)
Change of confining pressure = change of fluid pressure
Constant fluid pressure
Fig. 1- Schematicof unjacketedand jacketed compressibility test.
519