ISSN 1069-3513, Izvestiya, Physics of the Solid Earth, 2007, Vol. 43, No. 1, pp. 67–74. © Pleiades Publishing, Ltd., 2007.
Original Russian Text © V.M. Yarushina, Yu.Yu. Podladchikov, 2007, published in Fizika Zemli, 2007, No. 1, pp. 71–79.
The Effect of Nonhydrostaticity on Elastoplastic
Compaction and Decompaction
V. M. Yarushinaa,b and Yu. Yu. Podladchikova
a University
b Institute
of Oslo, Oslo, Norway
for Automation and Control Processes, Far East Division, Russian Academy of Sciences, Vladivostok, Russia
Received June 8, 2006
Abstract—The paper is devoted to the modeling of processes of inelastic nonhydrostatic compaction and
decompaction of low porosity geomaterials such that each pore can be considered as an isolated cavity in a solid
matrix. A real material containing a large number of noninteracting pores is replaced at a microscopic level by
an equivalent homogeneous medium containing a single pore inclusion. Attention is paid to the case where
cylindrical pores oriented perpendicularly to an applied nonhydrostatic load are present in the material. Along
with the elastoplastic response of the medium, viscous effects are considered. The effect of shear stresses on
compaction and decompaction of porous geomaterials is studied. The dependence of effective material parameters on the properties of components is examined.
PACS numbers: 91.32.De
DOI: 10.1134/S1069351307010077
INTRODUCTION
of shear stress required for the onset of compaction
induced by shear loads. Although it reproduces well
data on the critical pressure in the hydrostatic case, the
model fails to describe data of shear stress measurements.
This divergence of the model with experimental
data is possibly due to the fact that a hydrostatic stress
approximates somewhat inadequately the actual stress
state around pores [Curran and Carroll, 1979]. In the
nonhydrostatic case, additional problems can arise due
to the oversimplification associated with the assumption that the stress distribution around a pore is spherically symmetric [Baud et al., 2000]. External loads
applied to porous material concentrate stresses in the
vicinity of a spherical pore; as a result, a plastic flow
and residual strains arise in such zones. In order to
attain better agreement with laboratory data, the spherically symmetric model needs to be modified in a way
accounting for the shear stress and compaction–decompaction produced by shear loads.
Pore spaces in rocks can become compacted or
dilate in response to an applied load or a change in the
pore pressure. Traditionally, the compaction and settlement of an aquifer and reservoirs is analyzed in terms
of a poroelastic model [Biot, 1941; Skempton, 1954].
However, the analysis of compressive collapse of
porosity and generation of new porosity in many sedimentary, igneous, and metamorphic rocks is related to
fundamental ideas of inelastic behavior and processes
of porous rock fracture. The inelastic response of heterogeneous materials depends on both microstructural
characteristics (the size, shape, and orientation of
grains, inclusions, pores, and cracks) and mechanical
properties of components. Attempts to describe a
medium with a microstructure have been repeatedly
made by various authors [Lyakhovsky and Myasnikov,
1984; Myasnikov and Guzev, 1999; Yarushina, 2005].
The approach based on a spherical model [Mackenzie, 1950; Carroll, 1980] proved rather successful in
describing linear and nonlinear properties of porous
media. According to this approach, an isolated thick
spherical shell is regarded as the simplest model of a
medium with small porosity containing noninteracting
pores of a relatively small size. This model accounted
for the main features of the compaction process in
porous materials and proved effective for deriving an
equation of pore collapse agreeing well with experimental data. However, the spherical model of pore collapse has its own internal limitations because it fails to
reconstruct the yield surface for both hydrostatic and
nonhydrostatic loads with the same set of yield parameters. The model systematically overestimates the value
FORMULATION OF THE PROBLEM
We examine an inelastic response of low porosity
material containing noninteracting isolated pores and
subjected to an external nonhydrostatic load under the
conditions of plane deformation. The pores are
assumed to be cylindrical and oriented perpendicularly
to the direction of the external load. As a representative
volume element, we choose a hollow cylinder in an infinite medium (Fig. 1). We assume that external quasistatic loads applied to the macrovolume lead to similar
loading of a microvolume containing a single pore.
67
68
YARUSHINA, PODLADCHIKOV
y
the stress tensor components σij at the inner and outer
boundaries are
σy∞
σ rr
r=R
σ xx
σx∞
L
σ yy
p
x
σx∞
σy∞
Fig. 1. Model of a representative microvolume of a porous
medium. The dark area is the zone of the plastic deformation development.
Thus, the effect of the external medium on the given
volume reduces to compression in the compaction case
(Fig. 1) and, in the case of decompaction, to extension
by a couple of forces differing in absolute value and oriented in the horizontal and vertical directions, respectively. Given a sufficiently large value of these forces, a
plastic flow zone arises in the vicinity of the pore. Outside this zone, the material, as before, is in an elastic or
a viscous state. A constant normal pressure is applied at
the inner boundary of the pore in water-saturated
porous material. In the case of a dry rock, the inner
boundary is not loaded.
= σ y = const,
r=∞
= 0.
σ xx + σ yy = 4ReΦ ( z ),
(3)
σ yy – σ xx + 2iσ xy = 2 [ zΦ' ( z ) + Ψ ( z ) ].
We find, from boundary conditions (2)
2 ∞
∞
R τ
p
Φ ( z ) = ------ + -----------,
2
2
z
4 ∞
∞
R ( p – p ) 3R τ
∞
-,
+ -------------Ψ ( z ) = τ + -------------------------4
2
z
z
2
where
∞
∞
∞
σy – σx
∞
-.
τ = -----------------2
The values p∞ and τ∞ are, respectively, the pressure and
shear stress at infinity. The stress components in the
polar coordinates have the form
∞
σ rr = p – ∆P ( R/r )
∞
2
– τ ( 1 – 4 ( R/r ) + 3 ( R/r ) ) cos 2θ,
2
∞
∂σ
1 ∂σ rθ σ rr – σ θθ
+ --------------------- = 0,
---------rr- + --- ---------r ∂θ
r
∂r
σ rθ
∂σ rθ 1 ∂σ θθ
- = 0.
---------- + --- ----------- + 2 -----r ∂θ
r
∂r
σ xy
(2)
Here, (r, θ) is the polar system of coordinates used
together with the Cartesian coordinates (x, y) and R is
the radius of the pore hole.
If the external load is not overly high, the material
exhibits elastic behavior and the stresses and displacements can be found by the method of Muskhelishvili
[1966] for a plane problem. Introducing the spatial
complex variables z = x + iy and z = x – iy, the elastic
stress tensor components can be connected with the
complex potentials Φ(z) and Ψ(z), according to
Muskhelishvili, as
∞
This formulation of the problem is well known in
the mathematical theory of ideal plasticity and, under
several restrictions imposed on the applied external
forces, has an elegant analytical solution, obtained by
Galin [1946]. Thus, we assume that a plane deformation state is valid throughout the material. Then, the
equations of equilibrium have the form
= 0,
= σ x = const,
∞
r=∞
r=R
∞
r=∞
σx + σy
∞
-,
p = -----------------2
ANALYTICAL SOLUTION
σ rθ
= p = const,
4
∞
σ θθ = p + ∆P ( R/r ) + τ ( 1 + 3 ( R/r ) ) cos 2θ,
2
∞
(1)
We assume that the compression (extension) of the
layer is produced by a constant external load such that
4
(4)
σ rθ = τ ( 1 + 2 ( R/r ) – 3 ( R/r ) ) sin 2θ.
2
4
Here, ∆P = p∞ – p is the difference of pressures at infinity and inside the pore. Figure 2 shows the corresponding distributions of the maximum tangential stress
(Fig. 2a) and the average pressure (Fig. 2b) and the displacement field (Fig. 2c).
IZVESTIYA, PHYSICS OF THE SOLID EARTH
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THE EFFECT OF NONHYDROSTATICITY ON ELASTOPLASTIC COMPACTION
(a)
3.0
(b)
69
(c)
2.5
0.1
2.0
80
0.
0.7
5
1.5
0.
70
1.0
0.2
0.9
85
0
0
0.4
0.7
0.9
0
0.
0.3
0.5
1
2
3
0
1
2
3
0
1
2
3
Fig. 2. Elastic distribution of the maximum tangential stress (a), average pressure (b), and displacement field (c) around a circular
hole in the plate.
using the a priori unknown conformal mapping z = w(ς);
as a result, we obtain
If the Tresca or the Mises criterion
( σ rr – σ θθ ) + 4σ rθ = 4k ,
2
2
2
(5)
σ xx + σ yy = σ rr + σ θθ = 4ReΦ̃ ( ς ),
where k is the shear yield point, is taken as a yield criterion, the elastic solution will hold as long as the external load parameters meet the condition
∞
2 τ + ∆P < k.
σ yy – σ xx + 2iσ xy = ( σ θθ – σ rr + 2iσ rθ )e
where
Φ̃ ( ς ) = Φ ( w ( ς ) ),
kξ w̃ ( ς )
Ψ̃ ( ς ) = ----- ------------- ,
ς w' ( ς )
w ( ς ) = c ( ς + m/ς ), w̃ ( ς ) = c ( mς + 1/ς ),
(9)
∆P /k – 1
c = R exp ⎛ ------------------------⎞ ,
⎝
⎠
2
(7)
σ rθ = 0,
∞
where ξ = sgn ( ∆P ) , i.e., the sign of the difference of
the internal and external pressures applied to the
medium.
To find the stresses in the elastic region, we can use
relations (3) of the Muskhelishvili method. The functions Φ(z) and Ψ(z) should be determined from the
boundary conditions at infinity and from the condition
of continuity of stresses at the boundary between the
elastic and plastic zones [Kachanov, 1969]. However,
this boundary is unknown and is itself to be determined.
Following Galin, we map the elastic region external
with respect to the unknown elastic–plastic boundary L
onto the exterior of the unit circle γ (|ς| > 1) (Fig. 3),
IZVESTIYA, PHYSICS OF THE SOLID EARTH
Ψ̃ ( ς ) = Ψ ( w ( ς ) ),
w(ς)
p + kξ
Φ̃ ( ς ) = --------------- + kξ ln -----------,
ςR
2
σ rr = p + 2kξ ln ( r/R ),
σ θθ = p + 2kξ ( 1 + ln ( r/R ) ),
(8)
Φ̃' ( ς )
= 2 w ( ς ) ------------- + Ψ̃ ( ς ) ,
w' ( ς )
(6)
Plastic regions start developing in the material from the
moment when inequality (6) becomes invalid. These
regions can develop in two different ways: the plastic
region can encompass the entire circular hole, or both
plastic and elastic regions can coexist at the boundary
of the hole. Galin’s solution is valid for plastic regions
of the first type.
According to Galin, the stress distribution is, on the
strength of (1) and (2), centrally symmetric in the plastic zone encompassing the entire circular hole:
– 2θi
Vol. 43
τ
m = -----.
kξ
y
z
r
L
ζ
M
θ
x
ρ
z = ω(ζ)
M'
θ
γ
Fig. 3. Conformal mapping of the plate plane z onto the
complex plane ς.
No. 1
2007
70
YARUSHINA, PODLADCHIKOV
a 1
ln ⎛ ------ + ---⎞
⎝ 2b 2⎠
ln(b/R)
10
0.4
8
∆P = 2kτeff = 0.46
∆P = 10kτeff = 0.46
6
0.4
0.3
4
0.2
0.2 0.4 0.6 0.8 1.0
2
0.2 0.4 0.6 0.8 1.0
0
5
10
15
20
|∆P|/k
0.1
0
0.1
0.2
0.3
0.4
0.5
τeff
Fig. 4. Dependence of the sizes and elongation of the plastic zone on the external loading parameters τeff and ∆P.
Note that the value m is, in essence, the shear stress at
infinity reduced to the yield point. The elastic–plastic
boundary is the ellipse with the semiaxes a = c|1 + m|
and b = c|1 – m|. Its aspect ratio depends on the shear
stress at infinity τ∞, and its sizes depend on the pressure
∞
∞
difference ∆P (Fig. 4). If σ x = σ y , i.e., m = 0, which
corresponds to the case of a hydrostatic stress state, the
plasticity zone will have a circular shape. As the differ∞
∞
ence between σ x and σ y increases, the plasticity zone
will be increasingly more elongated. We have
Depending on the sign of the shear stress at infinity τ∞
and the sign of the pressure difference ∆P, the elliptic
plastic zone is oriented horizontally or vertically
(Fig. 5).
Figure 6a shows the maximum tangential stress
obtained from analytical solution (7)–(9), and the aver-
y
el
el
2µ ( u x + iu y ) = ( 3 – 4ν )ϕ ( ς )
w(ς)
– ------------- ϕ' ( ς ) – ψ ( ς ),
w' ( ς )
(10)
where µ is the shear modulus, ν is the Poisson ratio, and
∞
τ eff = – ln ( 1 – τ /k ).
m>0
age hydrostatic pressure is shown in Fig. 6b. The broken line is the elastic–plastic boundary. All stresses are
reduced to the yield point.
In the plane deformation case, the displacement
field in the elastic region is described by the relation
m<0
y
a
ϕ' ( ς ) = Φ̃ ( ς )w' ( ς ),
ψ' ( ς ) = Ψ̃ ( ς )w' ( ς ).
The full expression for the displacements is not written
out because it is cumbersome and can be found in
[Galin, 1946]. The displacement fields in the elastic
region are shown in Figs. 7a and 7b for the compaction
and decompaction cases, respectively. Approximate
solutions for the displacement fields in the plastic
region can be found, for example, in [Ivlev, 1957;
Erlikhman, 1970]. The given solution is also valid in
the case of a viscoplastic medium, with the only difference that relation (10) and Fig. 7 will represent the
velocity field.
b
pl
b
a
x
x
pl
Fig. 5. Dependence of the position of the plastic zone on the
sign of the ratio τ∞ and ∆P.
APPLICABILITY RANGE OF THE ANALYTICAL
SOLUTION
As mentioned above, the analytical solution of
Galin has several limitations. The first and major of
them is related to the assumption that the plastic zone
encompasses the entire contour of the hole. Moreover,
the plastic zone must be a doubly connected region and
all external forces applied to the volume under consideration must be constant.
IZVESTIYA, PHYSICS OF THE SOLID EARTH
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THE EFFECT OF NONHYDROSTATICITY ON ELASTOPLASTIC COMPACTION
(a)
0.2
71
(b)
0.4
0.6
0.8
1.0
1
2
3
4
Fig. 6. Elastoplastic distribution of the maximum tangential stress (a) and the average hydrostatic stress (b) obtained from Galin’s
solution.
(a)
(b)
8
6
4
2
0
–2
–4
–6
–8
–15
–10
–5
0
5
10
15
–15
–10
–5
0
5
10
15
Fig. 7. Displacement fields in the elastic region in the cases of compaction (a) and decompaction (b) derived from Galin’s solution.
In order for the plastic zone to completely cover the
circular hole, it is necessary that the circle of the radius
R lie inside the ellipse with the semiaxes a = c|1 + m|
and b = c|1 – m|. Thus, we have the following two
restraints on the parameters of the problem:
c 1 + m ≥ R⎫
⎬.
c 1–m ≥R⎭
The second limitation is related to the fact that new
plastic regions must not arise outside the elliptic plastic
zone covering the hole. This means that the following
condition must hold true everywhere outside the plastic
zone:
( σ rr – σ θθ ) + 4σ rθ ≤ 4k ;
2
2
IZVESTIYA, PHYSICS OF THE SOLID EARTH
(11)
Vol. 43
this inequality is valid only if the shear stress at infinity
satisfies the condition
∞
τ /k ≤ 0.4142,
(12)
following from solution (8) and inequality (11).
Figure 8 plots the dependence of the solution of problem (2) on the external loading parameters ∆P/k and
τeff. The first parameter is a dimensionless characteristic of the pressure difference at infinity and inside the
hole. The second parameter is the dimensionless effective tangential stress at infinity. The solution has different qualitative features in the five regions in Fig. 8.
Thus, zone I, shown in gray, is precisely the variation
range of parameters where Galin’s solution is valid.
Zone III corresponds to elastic solution (4) without
incipient regions of plastic flow. Plastic regions in
zones II and IV originate simultaneously in several sepNo. 1
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YARUSHINA, PODLADCHIKOV
τeff
IV
V
0.2 0.6 1.0
τeff = 0.9163, ∆P = 2k
0.2 0.6 1.0 1.0
τeff = 1.2040, ∆P = 4k
0.5348
0.2
0.6
1.0
τeff = 0.4308, ∆P = 1.1k
III
I
II
0.2 0.4 0.6 0.8
τeff = 0.4308, ∆P = 4k
0
1
2
3
|∆P|/k
4
Fig. 8. Dependence of the problem solution on the external loading parameters τeff and ∆P.
arate regions adjoining the hole contour. Zone V corresponds to the case where additional regions of plastic
flow (including shear bands) arise outside the elliptic
plastic zone described by Galin’s solution. Based on the
analytical solution, an approximate distribution of the
maximum tangential stress is shown for each of these
cases.
As seen from the diagram in Fig. 8, the analytical
solution holds only in a certain variation range of the
external loading parameters. Numerical modeling is
applied for detailed study of the behavior of the
medium under arbitrary external loads.
MODELING RESULTS
The most interesting problem in the theory of effective media [Mackenzie, 1950; Carroll, 1980; Mavko
et al., 2003] is the behavior of effective characteristics
of porous material at a macroscopic level (compressibility, elastic moduli, etc.) as a function of the properties of components and the presence and structure of
porosity. Traditionally, researchers focus on the pressure–volume strain interrelation, which can be obtained
from experimental data. We examine the relationship
between the applied pressure and the resulting change
in the microvolume. As an effective characteristic of the
microvolume change, we choose the value
εν =
°∫ u
L
n
⋅ dl/S,
(13)
representing, in the case of an elastic response of the
material, the flux of the displacement field across the
initial boundary of the cavity (pore) reduced to the initial area of its cross section. It is given by the expression
1–ν
1 – 2ν
el
ε ν = 2 ------------ ∆P + --------------- p.
µ
µ
(14)
Here, ν and µ are the elastic moduli of the solid phase
of the microvolume. If the solid phase is incompressible, the effective elastic volume strain of the microvolume depends linearly only on the difference ∆P
between the applied external pressure and the gas pressure inside the pore. In the case of dry material, the
effective elastic volume strain depends only on the
external pressure.
The effective volume strain behaves quite differently if plastic zones are present in the vicinity of the
cavity in the microvolume. Since an exact solution of
the elastoplastic problem does not exist for the displacement field in the plastic zone and the material
inside the plastic zone is supposed to be incompressible, εν in (13) will be meant as the flux of the displacement (velocity) field across the boundary of the plastic
zone reduced to the area of the zone. In the case of a
∞
∞
hydrostatic load ( σ x = σ y ), the effective volume strain
IZVESTIYA, PHYSICS OF THE SOLID EARTH
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THE EFFECT OF NONHYDROSTATICITY ON ELASTOPLASTIC COMPACTION
|∆P|
k
5.0
4.5
4.0
3.5
3.0
p2/k 2.5
2.0
p1/k 1.0
0.5
0
|∆P|
k
6
(a)
v = 0.5
73
(b)
0 ≤ v < 0.5
5
4
3
γ
2
p2/k
p1/k 1
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
ε1
ε2 |εv|
0
1
2
ε1 ε2
3
4
5
1 – 2ν
ε ν – --------------- p
µ
Fig. 9. Dependence of the volume strain on the difference between the pore and external pressures for incompressible (a) and compressible (b) matrices.
depends on the parameters of external loading at a
microlevel as
∆P
p kξ
pl
ε ν = ( 1 – 2ν ) ------- + ( 1 – 2ν ) --- + -----.
µ
µ µ
of a compressible solid phase. The following notation is
used in Fig. 9:
∞
k
τ
ε 1 = 2 ( 1 – ν ) --- ⎛ 1 – 2 --------⎞ ,
⎝
µ
k ⎠
(15)
∞
p
τ
-----1 = 1 – 2 -------- ,
k
k
In both elastic and elastoplastic cases of the hydrostatic load, we introduce an effective modulus of the
volume strain such that the increment εν is proportional
to ∆P/(3K); then the corresponding moduli will be
interrelated as
K
2(1 – ν)
------pl- = -------------------- .
1 – 2ν
K el
Vol. 43
∞ 2
(17)
∞
p
τ
-----2 = 1 – 2 ln ⎛ 1 – --------⎞ ,
⎝
k
k ⎠
(16)
Given nonhydrostatic external loading, the volume
strain of a microvolume should depend on the pressure
difference ∆P, the pore pressure p, and the external tangential stress τ∞. Figure 9 plots the dependence of the
volume strain of a microvolume on the difference
between the pore and external pressures for incompressible (Fig. 9a) and compressible (Fig. 9b) solid
phases. The first inclined segments in both plots correspond to the behavior of the microvolume before the
onset of plastic flow in it. The second inclined segment
reflects the elastoplastic (viscoplastic) behavior that is
not described by the analytical solution of Galin and
corresponds to region II in Fig. 8. Finally, the third segment corresponds to the elastoplastic (viscoplastic)
behavior described by Galin’s solution. As is evident
from Fig. 9a, a compaction limit exists in the case of the
elastoplastic behavior of a microvolume with an incompressible solid phase. Cylindrical pores can never close
completely. This theoretical result is supported by
experimental data [Uri et al., 2006]. Note that, as follows from Fig. 9b, this effect is not observed in the case
IZVESTIYA, PHYSICS OF THE SOLID EARTH
∞
k
3τ
τ
ε 2 ≈ 2 --- ( 1 – ν ) ⎛ 1 + --- -------- + ⎛ -----⎞ ⎞ ,
⎝ k⎠ ⎠
⎝
µ
2 k
µ
tan γ = ----------------------- .
k ( 1 – 2ν )
Formulas (17) show that, in the cases of both incompressible and compressible phases, the shear stress τ∞
makes a significant contribution to the inelastic variation in the effective volume strain (εν) of a microvolume. Moreover, the shear stress increases the value of
the volume compression. A similar effect in the behavior of porous geomaterials has been repeatedly
observed in experiments [Johnson and Green, 1976;
Curran and Carroll, 1979; Vajdova et al., 2004] and,
together with the dilatancy phenomenon observed in
several types of loose and porous rocks both before and
after the yield point [Lockner and Stanchits, 2002],
draws attention from researchers [Myasnikov et al.,
1990; Curran and Carroll, 1979].
We should note another interesting feature of the
medium under consideration. As seen from Fig. 9, the
macroscopic behavior of porous material, i.e., whether
local zones of plastic flow arise around the pores or the
No. 1
2007
74
YARUSHINA, PODLADCHIKOV
response of the material remains mostly elastic,
depends on the response of the majority of pores to the
applied external load. The pressure–volume compression diagram of the material will have different moduli.
The procedure of micromechanical averaging [Li and
Wang, 2004] can be used for determining at a macrolevel the constitutive relations of the porous material
under consideration.
CONCLUSIONS
Even in the simplest 2-D deformation model, the
incorporation of the elastoplastic mode of behavior of a
porous medium subjected to nonhydrostatic stress loading gives rise to such effects as shear enhanced compaction. Depending on the predominance of elastic or elastoplastic behavior in neighborhoods of pores, the
porous material will have different volumetric moduli.
In the case of a cylindrical geometry of pores and an
incompressible material of the solid matrix, the porous
medium has a compaction limit consisting in the fact
that the compaction becomes weaker with increasing
load. All these features cannot be discovered within the
framework of the elastic hydrostatic approach used in
the majority of studies.
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IZVESTIYA, PHYSICS OF THE SOLID EARTH
Vol. 43
No. 1
2007