Viscoplastic Interpretation of Localized Compaction Creep in Porous Rock

Abstract

Recent laboratory evidence shows that compaction creep in porous rocks may develop through stages of acceleration, especially if the material is susceptible to strain localization. This paper provides a mechanical interpretation of compaction creep based on viscoplasticity and nonlinear dynamics. For this purpose, a constitutive operator describing the evolution of compaction creep is defined to evaluate the spontaneous accumulation of pore collapse within an active compaction band. This strategy enables the determination of eigenvalues associated with the stability of the response which is able to differentiate decelerating from accelerating strain. This mathematical formalism was linked to a constitutive law able to simulate compaction localization. Material point simulations were then used to identify the region of the stress space where unstable compaction creep is expected, showing that accelerating strains correspond to pulses of inelastic strain rate. Such pulses were also found in full‐field numerical analyses of delayed compaction, revealing that they correspond to stages of inception and propagation of new bands across the volume of the simulated sample. These results illustrate the intimate relation between the spatial patterns of compaction and their temporal dynamics, showing that while homogeneous compaction develops with decaying rates of accumulation, localized compaction occurs through stages of accelerating deformation caused by the loss of strength taking place during the formation of a band. In addition, they provide a predictive modeling framework to simulate and explain the spatio‐temporal dynamics of compaction in porous sedimentary formations.

Full text

Viscoplastic Interpretation of Localized Compaction
Creep in Porous Rock
Ghassan Shahin1, Ferdinando Marinelli1,2 , and Giuseppe Buscarnera1
1Department of Civil and Environmental Engineering, Northwestern University, Evanston, IL, USA, 2Now at Plaxis BV,
A Bentley System Company, Delft, The Netherlands
Abstract Recent laboratory evidence shows that compaction creep in porous rocks may develop
through stages of acceleration, especially if the material is susceptible to strain localization. This paper
provides a mechanical interpretation of compaction creep based on viscoplasticity and nonlinear
dynamics. For this purpose, a constitutive operator describing the evolution of compaction creep is defined
to evaluate the spontaneous accumulation of pore collapse within an active compaction band. This strategy
enables the determination of eigenvalues associated with the stability of the response which is able
to differentiate decelerating from accelerating strain. This mathematical formalism was linked to a
constitutive law able to simulate compaction localization. Material point simulations were then used
to identify the region of the stress space where unstable compaction creep is expected, showing that
accelerating strains correspond to pulses of inelastic strain rate. Such pulses were also found in full-field
numerical analyses of delayed compaction, revealing that they correspond to stages of inception and
propagation of new bands across the volume of the simulated sample. These results illustrate the intimate
relation between the spatial patterns of compaction and their temporal dynamics, showing that while
homogeneous compaction develops with decaying rates of accumulation, localized compaction occurs
through stages of accelerating deformation caused by the loss of strength taking place during the formation
of a band. In addition, they provide a predictive modeling framework to simulate and explain the
spatio-temporal dynamics of compaction in porous sedimentary formations.
1. Introduction
The time-dependent deformation of rocks is crucial for numerous geophysical and geoengineering prob-
lems, from the spatial propagation of tremors (Cruz-Atienza et al., 2018) to land subsidence (Hettema et al.,
2002; Hol et al., 2018), and underground CO2storage (Rass et al., 2017; Torabi et al., 2015). In the brittle
faulting regime, delayed microcracking is a major source of creep (Brantut et al., 2013). In this context, sub-
critical fracture growth is widely regarded as the main source of spontaneous crack extension, especially
if augmented by the adsorption of fluids at the surface of active microcracks (Atkinson, 1984; Rice, 1978).
This form of creep is typically susceptible to delayed localized failure characterized by strain acceleration,
also referred to as tertiary creep (Kranz & Scholz, 1977). Recent experimental evidence on porous rocks has
revealed that a similar phenomenology can be found also with reference to compaction processes (Heap
et al., 2015), especially if the deformation is accompanied by grain crushing, another process sensitive to
fluid adsorption (Zhang & Buscarnera, 2018). From a phenomenological standpoint, compaction occurs at
stress levels associated with a plastic cap (Zhang et al., 1990). As a result, macroscopic hardening and strain
homogeneity are expected, ruling out the existence of failure mechanisms associated with tertiary (acceler-
ating) creep. However, it is nowadays known that high porosity rocks loaded in the compaction regime can
exhibit strain localization mechanisms leading to the emergence of compaction bands (Olsson & Holcomb,
2000).
These structures consist of narrow deformation zones that form perpendicular to the maximum principal
stress (Mollema & Antonellini, 1996), with evident concentration of inelastic processes such as pore collapse
and grain crushing (Aydin et al., 2006). Compaction bands were first documented in aeolian sandstone for-
mations in Nevada (Eichhubl et al., 2010; Hill, 1989; Sternlof et al., 2006) and Utah (Mollema & Antonellini,
1996). More recently, similar structures have also been found in the Rhône Valley, France (Ballas et al., 2013),
and the Syrian arch, Middle East (Gajst et al., 2018). Laboratory specimens collected from the field have
RESEARCH ARTICLE
10.1029/2019JB017498
Key Points:
• Delayed compaction banding is
interpreted as a viscoplastic material
instability
• Stability indices predicting
impending unstable creep in
compacting rock have been proposed
• It has been shown that unstable
compaction creep involves transient
pulses of inelasticity both at local and
global levels
Supporting Information:
• Supporting Information S1
Correspondence to:
G. Buscarnera,
g-buscarnera@northwestern.edu
Received 4 FEB 2019
Accepted 2 JUL 2019
Accepted article online 5 JUL 2019
©2019. American Geophysical Union.
All Rights Reserved.
SHAHIN ET AL.
Published online 21 OCT 2019
Citation:
Shahin, G., Marinelli, F., & Buscarnera,
G. (2019). Viscoplastic interpretation of
localized compaction creep in porous
rock. Journal of Geophysical Research:
Solid Earth,124, 10,180–10,196. https://
doi.org/10.1029/2019JB017498
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Journal of Geophysical Research: Solid Earth 10.1029/2019JB017498
Figure 1. Creep test on Bleurswiller sandstone showing the development of time-dependent compaction with distinct
patterns of (a) decaying axial deformation rate associated with diffuse (nonlocalized) compaction, and (b) stages of
growing axial deformation rate (hatched zone) associated with the emergence of discrete compaction bands (data after;
Heap et al., 2015).
shown that the pore collapse resulting from compaction localization may lead to several orders of magnitude
of permeability loss (Baud et al., 2012; David et al., 2001; Deng et al., 2015; Zhu & Wong, 1997).
Deformation experiments conducted at constant strain rate have shown that compaction bands can develop
over a wide range of effective pressures (Baud et al., 2004; Charalampidou et al., 2011; Fortin et al., 2006;
Holcomb & Olsson, 2003; Olsson et al., 2002; Tembe et al., 2008), especially at stress levels typical of cap
plasticity (Wong et al., 1997). Such experiments distinguished two types of compaction bands, namely, pure
compaction bands, which form perpendicular to the compression stress, and shear-enhanced compaction
bands, which develop at an angle with respect to the direction of loading (Shahin et al., 2019). Several authors
suggested that the variability of these types of compaction bands, as well as the transition from one type
of band to another, is mainly governed by porosity and grain size distribution (Liu et al., 2015) as well as
that existing spatial heterogeneities may affect the patterns of band growth (Baud et al., 2015; Fortin et al.,
2006). The influence of microstructural attributes is often regarded as one of the reasons why compaction
bands develop in some sandstones but not others, citing grain sorting, cement debonding, grain crushability,
and pore collapse as concurrent factors (Cheung et al., 2012; Katsman et al., 2005; Wang et al., 2008; Wong
& Baud, 2012). Aside from microstructural arguments, the propagation of compaction bands across rock
specimens occurs in distinct patterns that vary from discrete isolated bands (Fortin et al., 2005), to thickening
bands (Papazoglou et al., 2017), or hybrid growth mechanisms (Wong et al., 2001).
The rich data sets available about the spatial patterns of compaction band formation and propagation
have been recently expanded by laboratory studies which focused on their temporal evolution under fixed
boundary conditions (Heap et al., 2015). Specifically, transient stages of accelerated deformation have been
observed in conjunction with the initiation and propagation of compaction zones during creep tests con-
ducted on samples of Bleurswiller sandstone, a porous rock known to exhibit compaction localization
(Fortin et al., 2005). The samples were subjected to a constant deviator stress and left to creep over time,
revealing growth of compaction zones. Most notably, distinct patterns of temporal evolution of the axial
deformation were found (Figure 1), with trends of decaying or growing deformation rate that were assessed
as stable and unstable, respectively.
Despite the intriguing implications of these findings in terms of long-term prediction of crustal deformations
and reservoir quality deterioration, the literature offers limited tools to explain them and replicate their pat-
terns through computational analyses. For example, compaction localization is widely interpreted through
the bifurcation theory, based on which critical values of hardening modulus associated with the vanishing
of the acoustic tensor can be defined at the onset of compaction bands (Issen & Rudnicki, 2000). However,
these methods are specific for inviscid constitutive laws, which are notoriously unable to reproduce creep.
By contrast, if viscoplastic constitutive laws are used to simulate the rate and time dependence of the rock
behavior, localization analyses cannot be conducted in the same form, in that a tangent constitutive tensor
between stress and strain increments is no longer defined (Needleman, 1988). In this paper, we circumvent
these limitations by providing an interpretation of stable and unstable compaction creep that reflects the
temporal dynamics underlying the viscoplastic rock behavior, but at the same time is also consistent with
the standard indices used to differentiate homogeneous and localized compaction. Specifically, reference
is made to the mathematical formalism introduced by Pisanò and Prisco (2016) to identify instabilities in
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viscoplastic solids. According to this approach, the delayed deformation of a viscoplastic material can be
expressed in terms of multiple ordinary differential equations (ODEs), each associated with a stress/strain
variable describing the evolution of the current state. From a mathematical standpoint, this methodology
bears resemblance with the study of Lyapunov stability in multivariable dynamic systems (Hale & Koçak,
2012). From a geomechanical standpoint, it is an effective platform to cope with kinematics constraints due
to the presence of pore fluids, as recently shown for the liquefaction of loose granular solids at variable
degrees of saturation (Marinelli & Buscarnera, 2018; Marinelli et al., 2018). Hereafter, it will be shown how
this formalism can be generalized to compaction creep in porous rocks, especially for problems character-
ized by a marked potential for compaction localization. For this purpose, a strain-hardening elastoplastic
model (Gens & Nova, 1993; Lagioia et al., 1996; Nova et al., 2003), previously used to reproduce compaction
localization in porous rocks (Buscarnera & Laverack, 2014; Das & Buscarnera, 2014), will be augmented to
viscoplastic form (Perzyna, 1966) to simulate compaction creep. Material point analyses will be used to illus-
trate from a mathematical standpoint the differences between stable and unstable creep. Finally, the model
will be used within finite element computations to discuss the intimate connection between the temporal
dynamics of creep and spatial patterns of compaction with reference to virtual specimens of porous rock.
2. Stability Criteria for Viscoplastic Solids
In this study, the time-dependent behavior of porous rocks will be modeled through the overstress approach
proposed by the pioneering work of Perzyna (Perzyna, 1966), which has been successfully used to reproduce
the rate-dependent characteristics of a wide range of granular solids (Di Prisco et al., 2000; Lazari et al.,
2018; Tommasi et al., 2000; Yin et al., 2010). In this constitutive framework, the viscoplastic strain rate, .
vp
i,
is computed through a viscous nucleus function , which depends on the violation of the plastic constraints
(i.e., on the distance between the stress state outside the elastic domain and the yield surface, often referred
to as overstress):
.
vp
i=Φ()g
i.(1)
In this equation, fand gare the yield surface and the plastic potential surfaces, respectively, and ij is the
current stress state.
Under suitable conditions, such constitutive framework can suppress the ill posedness associated with strain
localization and restore the uniqueness of the field equations (Needleman, 1988). However, material insta-
bility can still be found in the form of sharp variations in the temporal evolution of the material response.
In this context, Oka et al. (1994) formulated a strategy to detect the onset of instabilities by showing that
the inception of accelerating trend of behavior during creep (i.e., an unstable response) can be identified
by rearranging the constitutive equations into an ODE. More recently, Pisanò and Prisco (2016) generalized
this idea to multidimensional systems characterized by a wider set of kinematic constraints, thus provid-
ing a direct link between the inception of accelerating response and the instability indices derived through
controllability analyses (Buscarnera et al., 2011; Imposimato & Nova, 1998; Nova, 1994).
According to this approach, the constitutive response can be reformulated as follows:
.
X=AX+F(2)
where Xand .
Xare the rate and the acceleration of the selected response variables, respectively. Ais a
constitutive operator governing the evolution of the system, and Fis a forcing term associated with the
imposed controls. It can be readily shown that during stages of creep, the external forcing vanishes and the
spontaneous evolution of the system in proximity of a given stress state is uniquely controlled by the local
characteristics of A, which can be inspected by studying the corresponding set of eigenvalues. Specifically,
negative eigenvalues indicate local exponential decay of the response variables (i.e., a response character-
ized by a stable behavior), while positive eigenvalues will indicate an incipient accelerating growth of the
same variables (i.e., unstable response).
Time-dependent compaction localization can also be studied through this formalism. However, since
viscous behavior implies the lack of the tangent operator linking directly stress and strain rate, strain local-
ization studies cannot be conducted in the usual form based on standard bifurcation theory (Bésuelle &
Rudnicki, 2004). As a result, equation (2) is a useful tool to circumvent the problem, provided that the local
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static/kinematic conditions active within a deformation band formation can be defined. This is simple for
pure compaction bands, in that they involve a local strain jump characterized by uniaxial deformation (i.e.,
orthogonal to the band and characterized by lack of lateral extension/contraction; Issen & Rudnicki, 2000).
These conditions resemble the kinematic constraints applied during oedometric (radially constrained) com-
pression (Arroyo et al., 2005; Buscarnera & Laverack, 2014) and can be readily inspected at a material point
level by considering axisymmetric stress conditions. Specifically, a potential compaction band during creep
would involve a constant axial stress (i.e., .
a=0), as well as a constrained radial strain (i.e., .
r=0)as
controlled variables. For this multivariable system, the kinematics of the strain jump in the active zone of
localized compaction can be described through a system of ODE compatible with equation (2) in which the
axial strain and the radial stress define the vector of the response variable X(i.e., X=[ .
a,.
r]). As a result,
the underlying system of ODEs governing this particular uniaxial deformation process can be written as
follows:
a
2r=



−Φ
 HCB

1−Φ2g
2
r
+Φ 2g
ar
0−Φ
 HCB −2E
(1−)Φ2g
2
r




A
.
a
2.
r+F
F,(3)
where Eand indicate the Young's modulus and Poisson's ratio, respectively, while HCB represents the
instability index for the selected controlled conditions, defined as:
HCB =H−H+H
Φ.(4)
where H is a term dependent on the loading rate, His the hardening modulus, and Hrepresents the con-
trollability index derived by Buscarnera and Laverack (2014) in the framework of rate-independent plasticity
by enforcing the kinematic conditions typical of a pure compaction band:
H=−g
r
E
2(1−)

r
.(5)
The derivations leading to this system of ODEs are provided in the supporting information. It is worth
remarking that, in case of creep, the ODEs described in equation (3) can be rewritten in a compact form (i.e.,
.
X=AX) resulting from the lack of forcing (i.e., F≡0and F≡0, as well as H=0). As a result, due to the
lack of coupling between the response variables aand r, the local stability of the system can be evaluated
by tracking the sign of the eigenvalues of A, which correspond to the diagonal terms:
(A)→




a=−
Φ
 HCB
r=a−2E
(1−)Φ2g
2
r
(6)
The process is locally stable if both eigenvalues are negative (Pisanò & Prisco, 2016), thus implying that the
response variables exhibit a decaying trend. By contrast, if at least one of the two eigenvalues is non-negative,
the process can be regarded as unstable, in that it exhibits an accelerated evolution of the correspond-
ing response variable. Furthermore, by observing equation (6), it can be noticed that when the stress state
reaches the condition a=0, the eigenvalue r, related with the radial stress is still negative. This result
indicates that the loss of stability in terms of axial strain anticipates the loss of stability in terms of the radial
stress; therefore, the analysis of arepresents a sufficient condition to characterize the stability of the sys-
tem reported in equation (3). Accordingly, the initiation of positive axial strain acceleration coincides with
the vanishing of the eigenvalue a, which is fulfilled when the hardening modulus is equal to the control-
lability index H(i.e., when HCB =0). It is worth remarking that the vanishing of the instability index HCB
identifying the inception of compaction band propagation for viscoplastic and rate-independent systems
(Buscarnera & Laverack, 2014) coincides the analogous constitutive criterion obtained through the strain
localization theory (Rudnicki & Rice, 1975).
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3. Constitutive Model
The material selected for this study is the Bleurswiller sandstone, a porous rock known to develop com-
paction bands over a wide range of confinements (Fortin et al., 2005) and recently studied by Heap et al.
(2015) through triaxial creep experiments. In this context, Marinelli and Buscarnera (2015) proposed a
numerical model of this rock based on a strain-hardening elastoplastic constitutive law proposed by Nova
and coworkers (Gens & Nova, 1993; Nova et al., 2003), which involves two hardening variables and a nonas-
sociated flow rule able to replicate stress-dependent pore collapse. The structure of the model is inspired
by classical critical state plasticity laws such as Cam clay (Wood, 1990), in which a single yield surface
evolves homothetically upon plastic deformation. The expression proposed by Lagioia et al. (1996) was used
to describe both the yield surface and the plastic potential:

g=AK1h∕Ch
h·B−K2h∕Ch
hP−P
c=0(7a)
Ah=1+
K1hMh
(7b)
Bh=1+
K2hMh
(7c)
Ch=(1−h)(K1h−K2h)(7d)
with the functions K1h,2hdefined as
K1h∕2h=h1−h
21−h



1±


1−4h1−h
h1−h2



.(8)
pand qrepresent the mean and deviator stresses, respectively, while is the corresponding stress ratio,
defined as follows:
p=1
3iiq=3
2sisi=q
p+Pt
(9)
in which sij =ij −pij,ij is the Kronecker delta. Under axisymmetric compression, pand qare be specified
as follows:
p=1+23
3q=1−3,(10)
with 3the minimum principal stress. The variable Pcrepresents the hydrostatic yield stress and is defined
through an additive combination of the hardening variables Psand Pm(i.e., P
c=Ps+Pm+Pt, where Pt
indicates the tensile stress and is equal to Pt=kPm). The former variable incorporates the contribution
of the skeleton packing, while Pmmimics the intergranular cementation. The shape of the two surfaces
in equations (7) and (8) is controlled by the shape parameters, Mh,h, and h, for which the subscript h
makes reference to either the yield surface (h≡f) or the plastic potential (h≡g). These expressions provide
considerable flexibility to fit experimental data, in that distinct sets of shape parameters can be used to adjust
the degree of nonassociativity and the shape of the elastic domain. The evolution of the hardening variables
(i.e., Psand Pm) is governed by the following rules:
.
Ps=Ps
Bp
.
p
v(11)
.
Pm=−mPm(.
p
v+m
.
p
s),(12)
where Bp,mand mare material parameters. Psreproduces mechanisms associated with porosity changes
(thus incorporating hardening in compacting geomaterials and softening in dilative geomaterials). By con-
trast, Pmintroduces softening mechanisms due to both volumetric and deviatoric plastic strain. The interplay
between these variables leads to a homothetical contraction/expansion of the yield surface, which affects
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Journal of Geophysical Research: Solid Earth 10.1029/2019JB017498
the accuracy with which localization processes and pos-tlocalization deformations are captured. This con-
stitutive law can be used in its original rate-independent form, according to which the plastic strains are
obtained through a standard consistency condition, as follows:
.
p
i=Λ g
i
(13)
with being a non-negative plastic multiplier. However, a simple adaptation of the model to viscoplas-
tic form allows the simulation of rate dependency and creep. For this purpose, the inelastic strain rate
in equation (13) can be replaced by the following viscoplastic flow rule consistent with the formalism in
equation (1):
Φ( )=
Pco (14)
where Pco is the initial hydrostatic yielding stress, the symbol •indicates the McCauley brackets, and is
a fluidity parameter (i.e., the inverse of viscosity). Isotropic linear elasticity is here employed to compute the
increment of the stress state .
ithrough the standard relation:
.
i=De
ikl
.
kle(15)
where e
kl is the elastic strain tensor and De
ikl is the isotropic linear elastic tensor.
3.1. Model Calibration
To simulate the mechanical behavior of Bleurswiller sandstone, reference will be made to the calibration
procedure proposed by Marinelli and Buscarnera (2015), which was based on the experimental data reported
by Fortin et al., (2005); see the supporting information for a detailed list of model parameters). Such proce-
dure relied on the strain localization theory, thus identifying the conditions at material point level that are
conductive of bifurcated solutions in the form of narrow deformation bands. For elastoplastic models, such
bifurcation criterion is written as follows:
A()=det n()Dep
iklnl()≤0,(16)
where represents the angle between the normal to the plane of the band and the direction of the maximum
compressive stress while Dep
ikl is the elastoplastic constitutive tensor. The study of the bifurcation criterion
reported in equation (17) (i.e., the evaluation of the stress states at which the function A()vanishes) can
be used to characterize the kinematics of a particular failure mode associated with a selected stress path.
Specifically, the pressure-dependent characteristics related to the band inclination can be assessed on the
basis of the angle min corresponding to the minimum of the function A()(i.e., A(min)=min(A)).
An example of this strategy is illustrated in Figures 2a and 2b in which the stress states predicting localized
compaction in the form of horizontal bands (i.e., min =0) have been plotted at initial yielding with a solid
red line, thus identifying the so-called localization domain. In particular, a transition zone between pure
compaction bands (i.e., min =0) and shear-enhanced compaction bands (i.e., min >0) can be observed
in Figure 2d, thus emphasizing the possible coexistence of different strain localization mechanisms. From
Figure 2 it is readily apparent that the bifurcation conditions are satisfied over a wide range of confinements,
spanning from 40 to 112 MPa, consistently with the experimental evidence reported in Figure 2 (i.e., data
points in Figure 2a), showing the stress states at which compaction banding was observed.
These results, here obtained by employing the localization theory (Rudnicki & Rice, 1975), can be recov-
ered also by using controllability indices (Imposimato & Nova, 1998; Nova, 1994) adapted for the control
conditions characterizing the kinematics of a compaction band. Specifically, as shown by Buscarnera and
Laverack (2014), it is possible to make reference to the index HCB =H−Hassociated with pure compaction
bands (Figure 2d). The performance of the model is further illustrated with reference to a set of simulated
triaxial compression tests. Figures 3a and 3b shows the predicted response under 40-, 60-, 80-, and 100-MPa
confinement pressure. The comparison against corresponding measurements illustrates a satisfactory agree-
ment between model simulations and experimental data. As a result, this constitutive law will be used in the
following to discuss the temporal development of compaction bands in light of a viscoplastic enhancement
of the constitutive equations.
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Journal of Geophysical Research: Solid Earth 10.1029/2019JB017498
Figure 2. Compaction band analysis for Bleurswiller sandstone. (a) Yield surface and measured yielding points at
which compaction bands were detected; the thick line indicates the predicted domain of localized compaction. (b, c)
Localization profile at two different stress states (marked as 1 and 2 in panel a), showing the evolution of the acoustic
tensor determinant as a function of the band angle. (d) Computed instability index showing compatibility with
predictions based on the bifurcation theory (data after Fortin at al., 2005).
4. Interpretation of Compaction Creep
To illustrate the connection between the phenomenology of compaction creep and concepts of viscoplastic
instability, a particular stress path has been computed at the material point level. Such stress path consists
of two stages: (i) a triaxial loading path used to bring the stress state in the proximity of the initial yield sur-
face and (ii) a creep stage performed to analyze the evolution of the strain over time, in which the control
Figure 3. Comparison between (a) experiments and (b) simulations for Bleurswiller sandstone (data after Fortin et al.,
2005).
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Figure 4. Material point analyses of constant-stress axisymmetric creep tests preceded by triaxial loading. (a) Path P1
with initial confinement of 80 MPa, intersecting the yield surface inside the compaction localization domain . (b) Path
P2with initial confinement of 115 MPa, intersecting the yield surface outside the localization zone. (c) Axial strain
evolution during creep stage for both paths. The first path displays a transition from accelerated to decelerated axial
strain rate (the point of transition is marked by a hollow square); the second path displays a decaying strain rate pattern
over the entire simulation. (d) Evolution of deviatoric stress. (d) Mobilized overstress (P1exhibits a peak , whereas P2
displays a decaying trend). (e) Variation of the instability index HCB for both paths, where the first path is associated
with initially negative value indicating unstable response. The second path displays positive index indicating stable
response. Tindicates current creep time, while Tc is the total creep time.
conditions were switched to radially constrained deformation in order to replicate the kinematics of a com-
paction band. To investigate the material response resulting from different initial conditions, two values of
initial confinement have been selected, namely, P1=80 MPa (at which the creep phase starts from a stress
state within the localization domain) and P2=115 MPa (at which the creep stage is initiated outside the
domain of potential compaction banding). The results of the analyses are illustrated in Figures 4a and 4b,
and they show two end-members of creep behavior, which can be explained in light of the theory presented
in the previous section.
Specifically, although in both cases the creep stage involved the expansion of the elastic domain, the relative
movement between the stress state and the yield surface varies from one case to another. For example, while
path P2was associated with a similar evolution rate for both yield surface and stress state, path P1exhibited
a much faster alteration of the stress state compared to the evolution of the yield surface (i.e., the viscoplastic
strain rate did not produce sufficient hardening to follow the local stress state evolution). The evolution of
the axial strain is depicted in Figure 4c, plotted as a function of the normalized time (i.e., the ratio between
the actual elapsed time and the whole duration of the creep simulation). It can be seen that each stress path
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Figure 5. Identification of stress region of unstable creep. (a) Stress paths used to identify unstable states (shaded
region). Triaxial probes at various confinement pressures are integrated and the state is marked unstable if the
corresponding index is negative. (b) Depiction of the location of the stress paths P1and P2with respect to the detected
instability zone. Path P1crosses the yield surface within the unstable zone, whereas path P2is always located outside
the shaded area.
is associated with a distinct pattern of axial strain evolution. P1is characterized by an initially growing rate
of strain accumulation, followed by a decaying trend of the axial strain rate, until the deformation stabilized
to 0.05. By contrast, P2is associated with a response characterized entirely by a decaying trend of the axial
strain rate until reaching 0.0008 axial strain. The evolution of the radial stress state is reported in Figure 4d.
The computations show that in path P1the stage of growing axial strain rate corresponds to a rapid change
of the stress state. By the end of this phase (i.e., when the deformation response stabilizes), the stress also
stabilizes. The latter trend is also found in path P2, where both axial strain and stress state evolve in a similar
fashion. As mentioned above, the adopted viscoplastic model allows the stress state to lay outside the yield
surface. This violation controls the magnitude of the overstress, , and in turn the viscoplastic strain rate. As
a result, the temporal dynamics of creep (i.e., whether the strain accelerates or decelerates) depends on the
relative movement of the stress state and yield surface, which are in turn controlled by the static/kinematic
constraints (i.e., the strain jump) and the hardening characteristics of the material, respectively. Figure 4e
provides the evolution of the computed overstress for both cases. It is readily apparent that the growing rate
of axial strain in path P1corresponds to a simultaneous pulse of overstress. From this figure it can also be
noted that the peak of matches with the transition point between accelerating and decelerating strain
rate (square symbol in Figure 4c). By contrast, the evolution of the axial strain associated with path P2is
characterized by a decaying trend. The overstress was also decaying with no pulses detected throughout the
simulation. These distinct patterns stem from differences in stability index that can be framed within the
context of the theory summarized in the previous sections. Figure 4f reports the evolution of the stability
index, HCB, for each stress path. It can be noticed that for P1,HCB is initially characterized by negative values,
thus signaling positive eigenvalues and tendency for strain acceleration. The point of transition and the
overstress peak both correspond to a change in sign of this instability index. Specifically, the decaying trend
observed in the evolution of the axial strain and overstress corresponds to a positive instability index. By
contrast, the evolution of this index in path P2is always characterized by a positive value (i.e., negative
eigenvalues) over the entire duration of the creep simulation, thus explaining the stable evolution of the
response variables.
The region of the stress path associated with accelerating compaction creep can be identified by replicating
triaxial stress simulations at multiple initial confinements, while the sign of the associated instability index
is assessed throughout the stress path (Figure 5a). Paths, P1and P2, are displayed along with the identified
instability zone in the stress space (Figure 5b). It can be readily seen that during compaction creep, path
P1intersects the yield surface within the unstable zone and remains within it until reaching its boundary.
By contrast, path P2intersects the yield surface outside the unstable zone and implies a continuously stable
stress evolution.
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Figure 6. Finite element simulation of constant-stress test compared against a simulation of imposed deformation at
constant strain rate. (a) Response of the specimen for constant-traction and constant-deformation-rate simulations.
(b) Temporal evolution of vertical deformation normalized by the specimen height. The shadowed domain in panels (a)
and (b) indicates the creep stage. (c) Cumulative volumetric plastic strain field of the constant-strain-rate simulation
(indicated by the letter Lin panel a). (d) Cumulative volumetric plastic strain field of the constant-traction test
simulation shown at various intervals (indicated by the letter Cin panel a). A comparison of the strain fields, in panels
(c) and (d), reveals that in both cases compaction propagates initially in a series of discrete bands, followed by a
propagation in the form of band thickening.
5. Full-Field Simulation of Creep Experiments
Material point analyses describe idealized homogeneous processes that should be regarded only as approxi-
mations of real experiments. Hereafter, a numerical simulation based on the finite element method is used to
illustrate the spatiotemporal richness of localized compaction creep, as well as to explain the full-field behav-
ior of rock samples in light of the viscoplastic theory discussed previously. The numerical test is conducted
with reference to a cylindrical specimen with aspect ratio height/diameter = 2. The purpose of the analysis
is to replicate the testing procedure discussed by Heap et al. (2015) in a controlled numerical setting. The
specimen is discretized by a finite element mesh consisting of more than 75,000 four-node tetrahedral linear
elements. A single element at the bottom of the specimen is assigned 98% of the hydrostatic yielding pressure
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Figure 7. Gauss point computations extracted from a full-field analysis based on constant-stress conditions. (a) The
stress path is shown together with the yield surface and instability zone, where the path violates the yield surface
at Point 1 and progresses in a quasi-axisymmetric path to Point 2. The stress state remains then almost unaltered up
to Point 3. (b) Temporal evolution of the axial strain. (c) Temporal evolution of the instability index HCB. (d) Temporal
evolution of the overstress, .
prescribed throughout the rest of the domain. This weak element promotes the triggering of localization
by altering the homogeneity of deformation field. The boundary conditions set at the base of the specimen
include fixed vertical displacements and a node with no horizontal translation to prevent rigid body motion.
The numerical specimen is then subjected to two-step loading paths consisting of a displacement-controlled
shearing followed by force-controlled creep. An initial confinement pressure of 80 MPa is imposed by apply-
ing on each boundary a normal pressure in equilibrium with an equivalent internal stress state. The radial
pressure is then kept constant and the nodes at the top boundary are translated vertically. The imposed axial
strain rate during the shearing stage is set to 1.0E−5s−1, in agreement with the experiments here used as a
reference. Once the desired stress level is attained, the traction at the top boundary is kept constant and the
specimen is left to creep over time. The response of the specimen is inspected from two standpoints. First,
the averaged vertical traction at the base (i.e., the reaction provided by the specimen to the imposed axial
deformation) is reported in Figure 6a. For the sake of comparison, the response of a numerical simulation
at constant displacement rate is also reported.
The vertical displacement at the top boundary is reported as a function of time in Figure 6b (continuous
line). In agreement with experiments on porous rocks susceptible to localized compaction creep (Heap et al.,
2015), the model reproduces the initial deceleration followed by an accelerating deformation response. The
transition between these regimes is reported in the same figure. These results clearly display stages of strain
acceleration reflecting an unstable deformation behavior and can therefore be considered consistent with
the material point simulations discussed previously.
To examine the spatial propagation of compaction emerging from constant-traction conditions, Figure 6d
shows the cumulative volumetric plastic strain field at several deformation intervals, noted by Ci, where i=
1−6. The figure reveals that the specimen exhibits a sequence of discrete bands, which spread sequentially
across the domain and span up to the fourth interval C4. Afterwards, compaction propagates in the form of
band thickening from the compacted zones at C4until the entire specimen is compacted. By comparing the
strain field evolution with its counterpart obtained from compression loading (Figure 6c-Li, where i=1−6.),
it can be readily seen that the propagation patterns reproduced by the two analyses are comparable, in that,
they are both characterized by the appearance of discrete bands. The predictions are also in part corroborated
by acoustic emission measurements documented by Heap et al. (2015), which reported the location of active
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Figure 8. Temporal evolution during the simulated creep test of (a) axial strain, (b) overstress, and (c) instability index
HCB at Gauss points located along the axis of the specimen. HCB = 0 is marked on all plots to show that the proposed
instability index is capable of identifying unstable creep.
compaction banding by identifying that multiple discrete bands formed across the specimen during the
initial stages of compaction propagation. Although the resolution of the measurements does not provide
conclusive evidence for the presence of a subsequent stage of band thickening, other tests conducted by
Fortin et al. (2009) on the same sandstone suggest a more widespread emission zone adjacent to discrete
bands, which may be consistent with the band thickening process predicted by the numerical model.
Despite the similarities between the simulation of localized compaction through imposed deformation and
creep stages, some important differences between these two types of simulations can also be found. For
example, the sequential formation of multiple bands during creep emerges at lower nominal strains com-
pared to those generated by compression (e.g., the appearance of two bands at stage L4occurs at a nominal
strain nearly twice than its closest counterpart in the creep simulation, i.e., stage C3, with this time difference
gradually increasing as more bands continue to appear). These differences can be attributed to variations
in the computed homogeneous strain field prior to compaction localization and suggest that spontaneous
localized compaction due to creep may occur at lower deformation thresholds than those emerging from
active compressive loading. Differences can be detected also in terms of the evolution of the computed
deformation patterns over time. Specifically, the spontaneous deformation under constant-traction required
longer intervals to develop multiple discrete bands. This is readily apparent by comparing the time at which
the two simulated samples are entirely covered by compaction bands, a stage which is expected to be fol-
lowed by rehardening. While upon compression, such condition is attained at t=2,400 min (stage L6),
the elapsed time at which a similar pattern is retrieved during creep is nearly twice (stage C6, computed at
t=4,500 min).
The connection between the globally unstable behavior and 6 local material instabilities can be inspected
on the basis of the material point interpretation presented in the previous section. For this purpose, Gauss
point computations at a randomly selected element is reported in Figure 7. The evolution of the stress state
is shown in Figure 7a plotted along with the numerically integrated instability zone in Figure 5a. It can be
noticed that the evolution of the stress path is characterized by features known to be signatures of uniaxial
deformation (i.e., q∕p=±2∕3). Furthermore, the evolution of the stress path was limited by the boundaries
of the instability zone. When the stress state crosses the outer boundary of the instability zone (i.e., at Point
2), the stress path changes direction, thus marking the deactivation of a previously active strain localization
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Figure 9. Full-field data from a constant-traction test simulation. (upper row) Incremental volumetric plastic strain
field confirming the activation of new discrete bands in the first three intervals followed by propagation of the already
activated zones. (lower row) Overstress field at the same intervals confirming the coincidence of overstress peaks with
active compaction bands.
zone. Kinematically, when viscoplasticity initiates at Point (1), the local response exhibits an acceleration
in the axial strain (Figure 7b). This acceleration continues across the entire stage between (1) and (2). This
unstable response is related to a negative sign of H−H, as can be seen in Figure 7c, and is associated with
a sharp increase in the overstress, which has a peak at the end of this stage (Figure 7c). Between (2) and (3)
the stress state remains almost unaltered, yet significant accumulation of axial strain develops, reflecting the
activation of delayed deformation inside the compaction zone. Later, the overstress diminishes, indicating
that after (3) there is no longer accumulation of further plastic strain. This analysis demonstrates that the
insights gained from the local viscoplastic interpretation of compaction creep are able to successfully depict
some key features of the dynamics of time-dependent compaction instability.
To further inspect the simulations, a number of elements at the central axis and over the entire height of
the specimen were selected, and the associated evolution of the axial strain, overstress ratio, and H−HXis
reported (Figure 8).
The point at which the instability index switches sign (i.e., H−HX=0) is marked in all plots. This result
confirms the above conclusions, in that, localized compaction is associated with an instability of the response
variable (i.e., axial strain) followed by a stable stage. It can also be seen that the point of transition from
unstable to stable response always corresponds to a peak of the computed overstress value (i.e., a spike
of viscoplastic strain rate). This result can be corroborated by inspecting the spatial field of incremental
variables. For example, Figure 9 shows the simultaneous evolution of the incremental volumetric strain field
and the current overstress field. The incremental strain field confirms the emergence of new active zones in
the first three intervals, and remarkable concentration of strain takes place in form of new bands and decays
over time. A similar pattern can be observed in the overstress field, indicating that the strain concentration
results from zones of high overstress.
6. Prediction of Compaction Band Growth for Different Boundary Conditions
While the previous section focused on conditions mimicking recent laboratory tests, hereafter the model is
employed to predict compaction band growth under boundary conditions not yet explored in experimental
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Figure 10. Full-field simulation of relaxation test compared with a creep test simulation. (a) Axial force evolution
reported as a function of time. (b) Volumetric plastic strain field taken at the time intervals marked in (a).
settings. Specifically, tests conducted at constant axial deformation (also referred to as relaxation tests) have
been simulated to assess changes in the spatiotemporal dynamic of localized compaction. Such conditions
are relevant for a variety of underground activities such as hydrocarbon reservoirs depletion (Pijnenburg
et al., 2018). In addition, they may provide useful insights to explore boundary conditions relevant for geo-
physical processes but rarely tested in conjunction with creep tests. This additional set of simulations is
based on the same specimen geometry discussed in previous sections. To facilitate visualization, the loca-
tion of the weak element is placed at the middle of the simulated specimen. The specimen is first subjected
to a given confinement pressure and then sheared by lowering the upper boundary at a constant rate (equiv-
alent to nominal axial strain of 1.0%) while maintaining the lateral stress constant. The shearing stage has
been defined such that a stress state able to initiate viscoplastic deformation is reached. The vertical velocity
of the upper boundary is then set to zero, and the specimen is left to relax. In order to compare the out-
come of fixed-displacement simulations with those of constant-traction conditions, the same simulation has
been repeated by replacing the last relaxation stage with a creep phase characterized by the same (constant)
nominal axial stress attained prior to the start of the relaxation path.
Figure 10a displays the mechanical responses associated with the relaxation test, where it can be seen that
the specimen exhibited sequential stages of axial stress loss. The first of such stages occurs right at the begin-
ning of relaxation (R1→R2) and results in a 10% loss of axial force carried by the simulated specimen.
Afterward, an additional stage of stress loss is obtained, resulting in an overall 40% decay of the sustained
axial force. Both stages of axial force decay are characterized by an initial sharp relaxation, followed by
stabilizing stress conditions.
Figure 10b shows snapshots of the volumetric strain field at various time intervals along the relaxation/creep
phase, from which it can be seen that both conditions lead to compaction localization. While compaction
localization during relaxation results into a single band growing at the middle of the specimen (i.e., exactly in
correspondence of the weak zone), constant-traction conditions lead to multiple bands scattered throughout
the sample volume. Such patterns are consistent with those shown in the previous section, thus emphasiz-
ing that creep simulations are not significantly affected by the location of the weak element. These results
indicate that the fixed boundaries inherent with relaxation conditions promote to the formation of isolated
deformation structures in proximity of existing flaws, as well as that the resulting force decay is sufficient
to shield much of the sample volume from inelasticity. By making reference to the axial stress loss, it can be
noticed that the first phase corresponds to the initial lateral growth of the band (R1→R2), while the second
phase of axial force decay corresponds to a stage of band thickening (R2→R5). These features indicate that
creep and relaxation tests display similar key stages of compaction band growth. In addition, they suggest
that creep may be suitable to study features such as band spacing, while relaxation may be convenient to
identify the geometric characteristics of a single band and isolate a dominant weakness zone.
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7. Conclusions
This paper discussed a mathematical approach based on nonlinear dynamics and viscoplasticity to inter-
pret the evolution of delayed compaction bands during stages of creep. For this purpose, the constitutive
equations have been recast in the form of an ODE specific for the kinematic conditions associated with the
strain jump within an active compaction band. In particular, an instability index connected with the eigen-
values of the constitutive operator governing the temporal dynamics during localized compaction creep was
derived and used to detect the transition from stable to unstable deformation in material point simulations.
The acceleration of the simulated deformation response variables was shown to be caused by a pulse of over-
stress related with the coordinated evolution of stress state and yield surface. This framework was shown
to be a useful tool to explain the emergence and propagation of compaction bands in full-field analyses of
compaction creep based on finite element simulations. An intimate relation between the spatial patterns
of compaction and their temporal progression was found, revealing that the pulses of overstress and vis-
coplastic strain rate concentrate at discrete locations. These spatial patterns were accompanied by a global
response characterized by an accelerating stage of deformation followed by a stage of decaying axial defor-
mation rate. While the accelerating stage was found to correspond to the formation of multiple compaction
bands, the decelerating stage indicated the homogeneous development of stable compaction throughout the
specimen, with no remaining pulses of overstress and localized strain acceleration. The framework of inter-
pretation discussed in this paper, as well as its use in numerical analyses of rock compaction, can provide
guidance to explain the mechanics of time-dependent compaction at laboratory and field scales. In particu-
lar, the possibility to define indices reflecting impending instability in the form of accelerating deformation
can be fruitfully used to predict the occurrence of catastrophic failure after extended stages of slow defor-
mation. While here only delayed compaction banding was discussed, this logic can in principle be extended
to a variety of failure mechanisms, such as distributed damage, frictional slip, dilation and shear bands,
thus potentially impacting a range of geophysical problems, simply by adapting the constitutive law for the
geomechanical problem at stake. It is therefore arguable that the techniques discussed in this work can be
used in the future to cope with a wide range of geophysical problems, such as the mechanics of seismogenic
faults, the onset of landslides and the collapse of volcanic edifices, and underground cavities.
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This research was supported by the U.S.
Department of Energy through grant
DE-SC0017615. The contents of this
work were exclusively of theoretical
and computational nature and did not
lead to the collection of new data. The
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