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ORIGINAL PAPER
Determination of the Effective Stress Law for Deformationin Coalbed Methane Reservoirs
Shimin Liu • Satya Harpalani
Received: 8 July 2012 / Accepted: 7 October 2013
� Springer-Verlag Wien 2013
Abstract Effective stress laws and their application are
not new, but are often overlooked or applied inappropri-
ately. The complexity of using a proper effective stress law
increases when analyzing stress variation in coal as a result
of gas production or mining. In this paper, an effective
stress law is derived analytically for coalbed methane
reservoirs, combining the concepts of matrix shrinkage/
swelling and external stress by including the effect of
sorbing gas pressure on the elastic response of the reser-
voir. The proposed law reduces to that of Terzaghi when
the compressibility of bulk material is sufficiently greater
than the compressibility of the solid grain, and without the
strain associated with matrix shrinkage/swelling effect.
Moreover, it is shown that the Biot coefficient (a) can have
a value larger than unity for self-swelling/dilation materi-
als, such as coal. The proposed stress–strain relationship
was validated using experimental results. Overall, the
effective stress law for deformation was extended for
sorptive materials, providing a new and unique technique
to analyze the elastic behavior of coal by reducing three
variables, namely, external stress, pore pressure and matrix
shrinkage/swelling along with the associated stress, down
to one variable, ‘‘effective stress’’.
Keywords Effective stress � Elastic deformation �Matrix shrinkage/swelling � Biot coefficient � Coal
1 Introduction
After a rather slow beginning in the 1980s, coalbed
methane (CBM) or coalbed gas has become an important
source of energy worldwide, currently accounting for
nearly 10 % of US annual gas production and approxi-
mately 12 % of the estimated total US natural gas reserves
(US EIA 2009). Significant activity is also underway in
Australia, Canada, China, India and Indonesia. For exam-
ple, the number of CBM wells in China increased from
near 0 in 2003 to 1,000 in 2007, with the corresponding
annual gas production increased to *1.8 million m3 in
2008 (Zhao 2009). Hence, it is believed that the contribu-
tion of CBM to the overall mix of natural gas resources
worldwide is going to significantly increase in the next
decade. This is further corroborated by the current switch
from the use of coal to natural gas for power generation.
It is well known that coal is a dual porosity gas reservoir
(Seidle et al. 1992; Harpalani and Chen 1995; Palmer and
Mansoori 1998; Ma et al. 2011). Coal is also well documented
as a self-swelling/dilating medium with sorbing gas(es) (Pan
and Connell 2007; Liu and Harpalani 2013a). Methane
transport mechanisms involved in production are typically
characterized by dewatering, gas desorption, diffusion and
viscous flow migration. These processes are restricted, as well
as influenced, by each other. A CBM reservoir is, therefore,
distinctly different from other conventional gas reservoirs.
The reservoir deforms as a result of fluids (water and gas)
depletion, which inadvertently changes the stress conditions
over the entire reservoir. The deformation is divided into three
categories, namely, the cleat (pore volume) deformation, coal
matrix linear elastic deformation due to stress variations and
coal matrix nonlinear elastic deformation due to the ‘‘matrix
shrinkage’’ effect induced by ad-/de-sorption of gas(es). The
term cleat is designated for coal which represents the
S. Liu (&)
Department of Energy and Mineral Engineering, Pennsylvania
State University, University Park, PA 16802, USA
e-mail: [email protected]
S. Harpalani
Department of Mining and Mineral Resources Engineering,
Southern Illinois University, Carbondale, IL 62901, USA
123
Rock Mech Rock Eng
DOI 10.1007/s00603-013-0492-6
macropore of coal; therefore, the macroporosity is usually
referred as cleat porosity. The conceptual coal deformation is
shown in Fig. 1. It is also well accepted that coal is a typical
porous medium and the effective stress law would, therefore,
link these three volumetric deformations associated with
depletion. Deformation of the coal reservoir is important
because it controls the permeability of coal during the
depletion process which, in turn, controls the long-term pro-
duction. To date, no study has been reported establishing an
effective stress law for the unique combination, typically
encountered in CBM reservoirs. It certainly is unique given
that desorption of gas results in matrix shrinkage strain, which
does not occur in traditional reservoir rocks. This phenome-
non is also believed to be responsible for the unusual pro-
duction trend exhibited by CBM reservoirs in the San Juan
basin. For example, the very first CBM well, Cahn in New
Mexico, went into production in 1981. More than 30 years
later, it continues to produce gas at a fairly high rates, certainly
beyond any expectations.
2 Literature Review
The formulation of the concept of effective stress is often
attributed to Terzaghi (1943). The original Terzaghi’s
effective stress formulation, known as the Terzaghi’s
Theory for one-dimensional consolidation, was given as
follows:
r0 ¼ r� p ð1Þ
where r0 is the effective stress, r is the total stress and p is
the pore pressure. This relationship was developed for soil
and is only representative of the particular case of saturated
soils with incompressible grains and a pore space com-
pletely filled with incompressible fluid. This formulation
has been shown to hold for soils for most practical appli-
cations and deviations from this expression are significant
and measurable only at extremely high pressures.
Subsequent to Terzaghi’s one-dimensional effective
stress, Biot (1941) extended this theory to three-dimen-
sional situations and presented the following effective
stress law:
r0ij ¼ rij � apdij ð2Þ
where r0ij is the effective stress tensor, rij is the total stress
tensor, p is the pore pressure, dij is the Kronecker symbol
and a is the effective stress coefficient, also commonly
referred to as the Biot coefficient. This is typically
considered a modified version of the Terzaghi effective
stress. It is also the more general and well-accepted form of
Fig. 1 Graphical representation of coal deformation. (a is the initial matrix dimension, a0 is the hypothetical matrix dimension assuming only
elastic deformation, a00 is hypothetical dimension assuming only matrix shrinkage due to desorption, b and b0 are cleat aperture)
S. Liu, S. Harpalani
123
effective stress for porous media. The theory is based on
isotropy of the material and linearity of the stress–strain
relationship. Biot (1941) proposed the following equation
to estimate the coefficient for soil consolidation:
a ¼ E
3ð1� 2mÞH ð3Þ
where E is the Young’s modulus, m is Poisson’s ratio and
introduced H is a new effective modulus. H is the solid
phase modulus of the soil grain, a measure of the com-
pressibility of soil particles. Biot (1955) extended the
consolidation theory to the more general case of anisotropy
and also argued that the effect of pore pressure should
actually be scaled down to weigh the respective reactions
proportionally to the volumetric fractions. Therefore, the
value of a was scaled down to the porosity of the material,
as a scaling factor for pore pressure (Biot 1955; Nuth and
Laloui 2008).
Skempton and Bishop (1954) presented a modification
of the effective stress law, expressing the Biot coefficient
as:
a ¼ 1� Rc ð4Þ
where Rc is the contact area between particles per unit
gross area of the material. However, in subsequent papers,
based on the same experimental evidence, both Bishop
(1955) and Skempton (1960) claimed that this formulation
cannot be correct and that the contact area between parti-
cles plays no role in formulation of the effective stress.
Almost four decades later, Li (2000) pointed out that it was
possible to use this coefficient in the estimation of the
structural effective stress, which controls the microstruc-
ture changes of the porous media.
Geertsma (1957) and Skempton (1960), on the basis of
their experimental data, suggested that:
a ¼ 1� K
Ks
or a ¼ 1� Cs
Cð5Þ
where K and Ks are the bulk modulus of dry porous
material and grain bulk modulus, respectively, and Cs and
C are compressibilities of the solid material (grains) and
skeleton. The ratio K/Ks (Cs/C) ranges from near zero to
unity based on experimental data, which depends on the
property of the porous media. Therefore, a ranges from
zero to one.
Another form of the Biot coefficient was proposed by
Suklje (1969) as:
a ¼ 1� ð1� /Þ K
Ks
ð6Þ
where / is porosity of the material. Since this derivation
lacked analytical rigor as well as data to back it up, it is
neither well known, nor did it find widespread acceptance.
A rigorous and theoretical derivation of Eq. (5) was
proposed by Nur and Byerlee (1971). Based on a series of
experimental investigations carried out using Weber
sandstone to determine the value of a, the authors showed
that the ratio of Cs/C was not negligibly small. Their pre-
dictions of effective stress using unity as the value of aproduced extremely low values, whereas the predictions
using Eq. (5) were excellent.
Lade and Boer (1997) proposed the general expression
for the effective stress in a porous media as:
a ¼ 1� Csku
Csks
ð7Þ
where Csku is the compressibility of the skeleton resulting
from a change in the pore pressure and Csks is the com-
pressibility of skeleton resulting from a change in the
confining stress. Based on experimental data, the authors
concluded that Terzaghi’s proposed effective stress prin-
ciple works well for stress magnitudes encountered in most
geotechnical applications, but significant deviations occur
at very high stresses/pressures.
Apart from the above studies, several empirical/experi-
mental studies have been completed to establish the value
of the Biot coefficient for different porous media. Fatt
(1959) measured a values ranging from 0.77 to 1.0 for
Boise sandstone using kerosene as the pore fluid and found
that its value depended on confining stress, with adecreasing as stress increased. For most practical purposes,
a value of 0.85 for a is used as a good approximation.
Hubbert and Rubey (1959a, b) and Knaap (1959) took the
stand that a = 1. Shiffman (1970) proposed that a should
be bounded by 1.0 and / (porosity), but did not explain
what actually controls the value of a between these two
boundaries. Christensen and Wang (1985) measured the
dynamic properties of Berea sandstone and presented avalues, obtained from deformation, ranging between 0.5 at
high stress/pressure and 0.89 at low stress/pressure. Zim-
merman et al. (1986) reported a & 1.0 based on experi-
mental data. Warpinski and Teufel (1992) evaluated the
deformation of tight sandstones and chalk in the laboratory
and reported that it varied with both confining stress and
internal pore pressure, ranging between 0.65 and 0.95.
Zhao et al. (2003) conducted an experimental study to
evaluate how the effective stress law for coal is affected by
methane and obtained the effective stress law. Based on a
large number of experimental results, the value of a was
regressed and shown not to have a constant value. It fol-
lowed a bilinear function of volumetric stress and pore
pressure. The authors also concluded that the pore pressure
had a larger impact on the value of a than the volumetric
stress and this was due to the strong influence of the con-
nectivity of pores and cracks in coal. The reported a value
ranged from near zero to near one.
Effective Stress Law for Deformation
123
Most recently, Izadi et al. (2011) analyzed the interac-
tion between fractures and coal matrix during coal defor-
mation. In their study, the role of swelling on the
deformation of coal was incorporated into the poro-elastic
continuum through an empirical Langmuir-type plot. The
authors also concluded that, in an elastic medium con-
taining swelling constituents, the relative change in pore
volume is of the same sense and in the same proportion as
the change in bulk volume. The pore and bulk volume
interactions controlled the evolution of flow behavior under
different boundary conditions.
3 Proposed Effective Stress Law for Coal Reservoirs
To summarize the work completed and reported, the Biot
coefficient (a) is a comprehensive representation of the
interaction between pore pressure and solid grain. The
variation of its value depends primarily on volumetric
stress of the rock mass (external stress), porosity of rock
(/), behavior of solid skeleton (compressibility of solid
grain and that of rock mass), consolidation type, the degree
of consolidation/cementation, etc.
For the purpose of CBM development and analysis, coal
is accepted as a dual porosity system, consisting of both
micropores and macropores. The physical structure of coal
is shown in Fig. 2. The face and butt cleats are the primary
and secondary avenues, respectively, for gas and water to
flow in coal. Both are commonly mutually orthogonal, or
nearly orthogonal, and are essentially perpendicular to the
bedding planes. For CBM reservoirs, the effective stress
law is of importance just as it is for conventional reservoir
engineering. It may quantitatively affect the coal reservoir
properties or processes, including the permeability,
deformation (subsidence), strength, pore volume (cleat)
compressibility, etc.
This paper is devoted entirely to deformation of coal as
a function of depletion, that is, continued production. Some
of the other important parameters/properties, such as the
strength, cleat compressibility and permeability will be
presented in subsequent papers. For this paper, the fol-
lowing basic properties of coal are assumed: (1) isotropy,
(2) reversibility of stress–strain relationship under final
equilibrium conditions, elastic behavior, (3) small overall
strains and (4) adsorbed phase gas that does not result in
additional pore pressure, which is an accepted assumption.
3.1 Definition of Effective Stress
The concept of ‘‘effective stress’’ has long been used in
both soil and rock mechanics. Terzaghi (1943) first pro-
posed the concept of effective stress in 1936. For him, the
term ‘‘effective’’ meant the calculated stress that was
effective in moving soil or causing displacements. Based
on this concept, he stated that the effective stress repre-
sented the average stress carried by the soil skeleton. The
Terzaghi effective stress solely focused on the elastic and
elasto-plastic constitutive equations of the solid skeleton of
the medium, which linked a change in stress to strain-like
quantity of the solid skeleton. This definition is somewhat
questionable since ‘‘the average stress of skeleton’’ is not
the only stress in the porous media; therefore, it cannot
fully determine the stress–strain behavior of the porous
media (soil).
In the current effort, the effective stress is defined as an
equivalent stress that controls the volumetric changes of a
porous sorptive medium, such as coal, based on a unique
combined effect of linear elastic and nonlinear elastic
deformations, and it links both external stress and pore
pressure to the strain behavior. The nonlinear elastic
deformation is defined for incorporating the sorption-
induced strain into the stress–strain relationship. It has been
experimentally and theoretically shown that the sorption-
induced strain is a nonlinear elastic deformation (Pan and
Connell 2007; Liu and Harpalani 2013a). Here, the concept
of effective stress converts a multi-phase medium with
multi-stress effects to a mechanically, single-phase, single-
stress continuum. It enters the elastic constitutive equations
of solid phase, linking a change in stress to that in strain.
The authors emphasize that all of the previous effective
stress relationships mentioned in the literature are also
applicable to elastic strain alone, just as proposed here.
This effective stress will not only significantly simplify the
calculation of the deformation of sorptive porous media,
but also provide a fundamental means to integrate the
sorption-induced deformation into the stress–strain rela-
tionship for sorptive porous media, such as coal.Fig. 2 Physical structure of coal containing cleats and micropores
(plan view)
S. Liu, S. Harpalani
123
3.2 Concept of Shrinkage/Swelling Strain–Stress
for Coal
For the purpose of CBM modeling, a bundle of matchstick
geometry (Seidle et al. 1992; Harpalani and Chen 1995;
Palmer and Mansoori 1998; Shi and Durucan 2004; Palmer
et al. 2007; Ma et al. 2011; Liu and Harpalani 2013b),
shown in Fig. 3, is well accepted. It consists of two sets of
vertical fractures, namely, the face and butt cleat (illus-
trated in Fig. 2), with blocks of coal matrix in between.
However, the idealized matchstick geometry of coal
oversimplifies the coal fracture system shown in Fig. 3
(Liu and Rutqvist 2010). Since the porosity of coal is fairly
low, compared to conventional reservoirs, the solid phase
matrix of coal must be in contact, at least partially, with
each other. Figure 4 shows a simplified vertical section of a
coal seam with cleats (fractures).
The shrinkage/swelling of coal as a result of gas
desorption/adsorption is a well-accepted phenomenon
(Levine 1996; Pan and Connell 2007; Liu and Harpalani
2013a, b). Since the reservoir is externally constrained by
lateral confinement and overlying strata, an additional
stress is imposed on the coal matrix blocks due to the
shrinkage/swelling strain. Here, we call it the ‘‘shrinkage/
swelling stress’’ and denote it as ‘‘rsm’’.
For the shrinkage/swelling strain during desorption/
adsorption, a theoretical model (Liu and Harpalani 2013a,
b) was developed and validated using data obtained in the
laboratory by the authors and available in the open litera-
ture (Moffat and Weale 1955; Levine 1996; Pan and
Connell 2007). The linear strain of coal due to de/ad-
sorption alone was shown to be:
el ¼aqRT
EAV0
Zp
0
b
1þ bpdp ¼ aqRT lnð1þ bpÞ
EAV0
ð8Þ
where el is the linear strain due to shrinkage/swelling, a and
b are Langmuir sorption constants, q is coal solid phase
density, R is the universal gas constant, T is the reservoir
temperature, EA is the modulus of solid expansion defined
by Maggs (1946), V0 is gas molar volume (22.4 m3/mol)
and p is the reservoir pressure. Since the shrinkage/
swelling strain is small, we assume that Hooke’s law is
applicable for the relationship between shrinkage/swelling
stress (rsm) and strain (el). The following stress–strain
relation was thus proposed:
rsm ¼ Eel ¼E
EA
� aqRT lnð1þ bpÞV0
ð9Þ
where E is the Young’s modulus of coal. Based on Eqs.
(8) and (9), the shrinkage/swelling stress and strain can
be easily calculated using geomechanical and thermal
parameters for different gas pressures. This shows that
rsm is a function of pore pressure and would, there-
fore, vary over the life of a CBM reservoir. Under
constant temperature (in situ) conditions, the higher the
adsorption capacity, the larger is the shrinkage/swelling
stress.
3.3 Elasticity of Coal Under CBM Reservoir
Conditions
For an isotropic porous medium, without shrinkage and
swelling, the general tensor expression for strain in terms
of stresses and pore pressure can be written as (Nur and
Byerlee 1971; Detournay and Cheng 1993):
Fig. 3 Bundle of matchstick geometry of coal showing flow through
vertical fractures
σv
σH σH
σsm
σsm
σsm
σsm
σsm
σsm
Fracture/Cleat
Matrix
Bedding Plane
σv
σH σH
σsm
σsm
σsm
σsm
σsm
σsm
Fracture/Cleat
Matrix
Bedding Plane
Fig. 4 Conceptual model for shrinkage/swelling stress in coal matrix
Effective Stress Law for Deformation
123
eij ¼1
2Grij �
rkk
3dij
� �þ 1
9Kdijrkk �
1
3Hdijp ð10Þ
where dij is the Kroenecker’s delta, eij is the small strain
tensor, rij is the Cauchy stress tensor, K and G are
identified as the bulk and shear modulus of drained elastic
solid, p is the pore pressure and H is solid phase modulus
(Biot 1941). Furthermore, the convention used is that
compression is positive. rkk is defined as:
rkk ¼ r11 þ r22 þ r33 ð11Þ
The isotropic compressive stress, denoted as P, is defined
as:
P ¼ 1
3rkk ¼
1
3r11 þ r22 þ r33ð Þ ð12Þ
It should be noted that the isotropic compressive stress is
also termed ‘‘mean stress’’ or ‘‘total pressure’’ in the
literature.
The first term on the right side of Eq. (10) is the strain
due to deviatoric stress, which depends only on the shear
modulus (G) of rock without any pore pressure. The second
term is the strain due to hydrostatic stress alone, which
depends only on the bulk modulus (K) of the rock without
pore pressure. The last term is the strain due to pore
pressure and depends on the effective modulus, H (Biot
1941).
For CBM reservoirs, there is an extra strain due to
desorption/adsorption-induced matrix strain. Using the
same concept as Biot (1941), the general tensor expression
for strain, in terms of the stresses and pore pressure, is thus
proposed as:
eij ¼1
2Grij �
rkk
3dij
� �þ 1
9Kdijrkk �
1
3Hdijpþ
1
3Zp
dijp
ð13Þ
where Zp is the effective shrinkage/swelling modulus. Zp
is not constant, like K and G, because the shrinkage/
swelling strain follows the Langmuir-type relationship
with pressure (Levine 1996; Harpalani and Mitra 2010;
Liu and Harpalani 2013a). Zp is, therefore, a function of
pore pressure.
Based on the stated assumptions and Eq. (13), it is
appropriate to state that pore pressure influences the nor-
mal, but not the shear strain. The normal strains of coal
with shrinkage/swelling effect are given as:
e11 ¼1
2Gr11 �
rkk
3
� �þ 1
9Krkk �
1
3Hpþ 1
3Zp
p ð14Þ
e22 ¼1
2Gr22 �
rkk
3
� �þ 1
9Krkk �
1
3Hpþ 1
3Zp
p ð15Þ
e33 ¼1
2Gr33 �
rkk
3
� �þ 1
9Krkk �
1
3Hpþ 1
3Zp
p ð16Þ
Then the volumetric strain can be calculated as:
ev ¼ e11 þ e22 þ e33 ¼1
KP� 1
Hpþ 1
Zp
p ð17Þ
The above equation can be simplified by introducing the
effective stress coefficient ‘‘a’’:
ev ¼1
KP� apð Þ ð18Þ
where a ¼ KH� K
Zp. Therefore, the volumetric strain can be
expressed as a function of a combination of pore pressure
and confining stress. The effective pressure, Pe, is now
defined as:
Pe ¼ P� ap ð19Þ
Equation (18) thus becomes:
ev ¼1
KPe ð20Þ
Equation (20) indicates that, by using the definition of
effective pressure shown in Eq. (19), the expression for
stress–strain in coal, which is a multi-phase medium with
multi-stress/pressure effects, reduces to a mechanically
equivalent, single-phase, single-stress continuum. It is also
concluded that effective stress is the link between the
elastic mechanics and poro-elastic mechanics. The poro-
elastic mechanics problems are thus significantly simplified
using the concept of effective stress. It also reduces the
complexity of numerically analyzing the pore pressure/
stress variation in a porous medium.
For more general conditions, the effective stress is
mathematically defined as:
reij ¼ rij � apdij ð21Þ
Based on the above effective stress tensor definition, the
strain tensor of coal can be written as:
eij ¼1
2Gre
ij �re
kk
3dij
� �þ 1
9Kdijr
ekk ð22Þ
The strain tensor was also given by Nur and Byerlee
(1971). This shows that the static problem of deformation
of coal with changes in pore pressure can be reduced to an
elastic problem in a non-porous material. However, the
value of a is still unknown. A theoretical derivation of a is
presented in the following section.
3.4 Theoretical Derivation of a
For a porous medium without shrinkage/swelling (Nur and
Byerlee 1971) provided a rigorous derivation, from basic
principles, for an exact expression for a. They then
obtained the corrected effective stress for elastic com-
pression. In their derivation, the state of stress for a porous
S. Liu, S. Harpalani
123
medium, subjected to a confining stress, P, and a uniform
pore pressure, p, was achieved conceptually in two steps.
First, they applied the pore pressure (p) and an equal
confining stress, that is P0 ¼ p. Second, they applied the
remaining confining stress, P00 ¼ P� p, without any fur-
ther change of pore pressure. This decomposition proce-
dure is illustrated in Fig. 5, where the principle of
superimposition is applied to calculate the volumetric
strain of the porous medium. The following volumetric
strain was obtained after rigorous derivation:
ev ¼1
KP� 1
K� 1
Ks
� �p ð23Þ
where Ks is the modulus of the solid grains.
For CBM reservoirs, there is an extra volumetric strain
induced by the sorption-associated shrinkage/swelling. The
volumetric strain is theoretically proposed by Liu and
Harpalani (2013a, b) as:
eav ¼
3aqRT
EAV0
Zp
0
b
1þ bpdp ¼ 3aqRT lnð1þ bpÞ
EAV0
ð24Þ
where eav is the volumetric strain induced by shrinkage/
swelling of coal. If the shrinkage/swelling effect of coal is
incorporated, the volumetric strain of the coal block is
given as follows by modifying Eq. (23):
ev ¼1
KP� 1
K� 1
Ks
� �p� 3aqRT lnð1þ bpÞ
EAV0pp ð25Þ
Re-arranging Eq. (25) gives the following:
ev ¼1
KP� 1� K
Ks
� �þ 3KaqRT lnð1þ bpÞ
EAV0p
� �p
�
ð26Þ
Comparing Eqs. (20) and (26), the effective stress can then
be written as:
Pe ¼ P� 1� K
Ks
� �þ 3KaqRT lnð1þ bpÞ
EAV0p
� �p ð27Þ
Therefore, the Biot coefficient is given as:
a ¼ 1� K
Ks
þ 3KaqRT lnð1þ bpÞEAV0p
ð28Þ
Every parameter in this expression can be measured/
estimated.
Fig. 5 Stress decomposition of
a representative porous element
(coal)
Effective Stress Law for Deformation
123
4 Experimental Validation
4.1 Specimen Preparation and Experimental Apparatus
To demonstrate the physical validity of the derived theo-
retical effective stress law for the coal deformation, a
laboratory-based study was carried out. In the experimental
work, we only measured strain under unconstrained
hydrostatic condition, since measurements under deviatoric
stress conditions are complicated and time-consuming.
These will, therefore, be provided in subsequent work.
The tested coal sample was obtained from the San Juan
basin basin, New Mexico (USA). The coal sample was kept
under water to prevent drying and oxidation until required
for testing. Detailed specimen preparation procedure is
provided in a prior publication (Liu and Harpalani 2013a).
The experimental apparatus for the unconstrained hydro-
static condition stress–strain measurement was designed to
enable us to measure the principal strains caused by gas
flooding. The volumetric strain was computed by adding up
all three measured principal strains. The schematic of the
experimental apparatus is shown in Fig. 6. The apparatus
consisted of high pressure vessels of capacity 15 MPa,
pressure monitoring and recording system, and a data
acquisition system for both pressures and strains.
4.2 Experimental Procedure
To compare the deformation behavior of coal in sorbing
and non-sorbing gas environments, helium and methane
were used as testing fluids. The specimen was first sub-
jected to increasing helium pressure in a step-wise manner.
For helium, known to be non-sorbing, the measured
principal strains were recorded as a result of changes in the
surrounding helium pressure. Helium was then bled out and
the specimen was subjected to methane flooding, in steps of
*1.38 MPa (200 psi), to a final pressure of *6.9 MPa
(1,000 psi). The corresponding principal strains were
continuously monitored and recorded. The measured
principal strains were used to calculate the volumetric
strains. The estimated volumetric strains were then used to
analyze the stress–strain behavior under unconstrained
hydrostatic condition.
4.3 Experimental Results and Analysis
The coal specimens were first dosed with helium to a
maximum pressure of *6.9 MPa (1,000 psi) in a step-wise
manner. The specimen was allowed to equilibrate by
ensuring that the strain remained stable for 1 day or more.
Fig. 6 Schematic of the
experimental apparatus used for
coal deformation
Fig. 7 Volumetric strain with increasing helium pressure
S. Liu, S. Harpalani
123
The volumetric strain was estimated by adding up the three
principal strains and was plotted versus gas pressure. The
results are shown in Fig. 7. As expected, the volume of
coal decreased with increasing helium pressure due to
mechanical compression of the solid coal grain; since there
was no deviatoric stress applied, the effective stress
(Eq. 19) was given as:
Pe ¼ ð1� aÞp ð29Þ
Since helium is a non-sorbing gas, the Biot coefficient
(Eq. 28) was reduced to the Geerstma (1957) form as in
Eq. (5). The stress–strain described in Eq. (20) then took
the following form:
ev ¼1
KPe ¼ p
K1� 1� K
Ks
� �� �¼ 1
Ks
p ð30Þ
Based on the effective stress law, this stress–strain
relationship with helium injection has been experimentally
confirmed by George and Barakat (2001). Therefore, the
solid phase bulk modulus (Ks) was estimated to be
Ks ¼ 11:97E�4 ¼ 5075 MPa. This value was used to validate
the methane injection experimental results.
After the helium cycle was completed, the specimen was
evacuated for a few days to release the residual helium.
The specimen was then flooded with methane to a final
pressure of 6.9 MPa (1,000 psi). Each injection step took a
fairly long period of time to attain equilibrium since
adsorption is a slow process. The principal strains were
monitored and recorded during the entire course of meth-
ane injection. Using the measured principal strains, the
volumetric strain of the coal specimen was calculated and
plotted as a function of pore pressure, as shown in Fig. 8.
These results were also reported in a recent publication
(Liu and Harpalani 2013a, b). With methane injection, the
Biot coefficient was as given in Eq. (28). Therefore, the
stress–strain relationship was obtained by substituting Eqs.
(28) and (29) into (20), as shown below:
eV ¼1
K
K
Ks
� 3KaqRT lnð1þ bpÞEAV0p
� �p
¼ p
Ks
� 3aqRT lnð1þ bpÞEAV0
ð31Þ
The input parameters are listed in Table 1, except Ks,
which was estimated using helium injection. A justification
of the input parameters is provided in a prior publication
(Liu and Harpalani 2013a). The predicted volumetric strain
along with the experimental data is presented in Fig. 9. The
modeled results predict the same trend of volumetric strain,
although the values are slightly higher than those mea-
sured. The proposed effective stress law can, therefore,
model both non-sorbing gas and sorbing gases with
acceptable accuracy. With the help of the proposed effec-
tive stress law, the stress–strain relationship was simplified
from poro-elastic to non-porous single-phase material. In
addition, it is a first attempt of its kind to incorporate the
sorption-induced strain into a single stress–strain relation-
ship. The technique provides a fundamental means for
deformation analysis of self-swelling/dilating material,
avoiding the complexity of poro-elastic and nonlinear
sorption effects.
5 Discussion
Coal is a unique porous medium exhibiting matrix
shrinkage/swelling behavior as a result of desorption/
adsorption of gas(es). The focus of this paper is on the
Fig. 8 Volumetric strain with increasing methane pressure Fig. 9 Modeled volumetric strain with methane pressure along with
measured data
Table 1 Input parameters for
the modeling workq(ton/m3)
R
(MPa*m3/kmol*K)
T
(K)
EA (MPa) V0
(kmol/m3)
a
(m3/t)
b
(MPa-1)
Ks
(MPa)
1.4 0.008314 308 1,600 22.4 19.1 0.4 5,075
Effective Stress Law for Deformation
123
stress–strain behavior. The Terzaghi effective stress law is
only valid for relatively loose material (soil), and it is
clearly inappropriate to use it to study the stress–strain
relationship of coal. Moreover, CBM reservoirs exhibit an
extra strain caused by desorption and adsorption. The thrust
of this paper is in developing an effective law for stress–
strain behavior of coal and validating with laboratory
results. A technique to estimate the Biot coefficient is also
provided, as given by Eq. (28).
Based on Eq. (28), a qualitative description of the
function a is presented in this paper. There are three pos-
sible scenarios depending on the magnitude of the coal
porosity and gas type.
1. Loose Material without Shrinkage/Swelling Effect:
The third term on the right side of Eq. (28) reduces to
zero. The second term is a function of porosity. If the
porosity is very high, the bulk modulus (K) is much
smaller than the intrinsic grain solid phase modulus, Ks
(K � Ks), which is often the case for loose material,
like soil, and a & 1. In other words, when the material
bulk compressibility is significantly greater than the
intrinsic compressibility of the solid grain, and the gas
does not adsorb on the solid material, a & 1 and the
effective stress Eq. (27) reduces to the following:
Pe ¼ P� p ð32Þ
which is the original effective stress for compression
and consolidation, as suggested by Terzaghi and dis-
cussed by Hubbert and Rubey (1959a, b), Skempton
(1960) and many others.
2. Tight Materials without Shrinkage/Swelling Effect:
The third term on the right side of Eq. (28) is zero.
However, the second term will not approach zero since
K is not significantly smaller than Ks. Since the coal
cleat porosity is fairly low, it can be considered a tight
material. The value of a is then given by Eq. (5). This
has been theoretically derived by Nur and Byerlee
(1971). The effective stress law reduces to the
following:
Pe ¼ P� 1� K
Ks
� �p ð33Þ
George and Barakat (2001) carried out experimental
investigations on the effective stress law using both
methane and helium. For helium, known to be a non-
sorbing gas, the value of a was estimated to be 0.71.
Here, the authors would like to point out that the ratio,
K/Ks, strongly depends on the porosity. For high
porosity, K is a small fraction of Ks. When the porosity
approaches zero, K & Ks and a approach zero, making
Pe ¼ P a single-phase solid body. This is as expected
from the theory of elasticity. In other words, the
porosity effect in the above effective stress expression
is not explicit; instead, it is included in the value of the
effective bulk modulus, K, of the material. Moreover,
the range of a is between 0 and 1.
3. Tight Material with Shrinkage/Swelling Effect (coal):
The Biot coefficient is given by Eq. (28). The value of
a is not only a function of coal properties, including
the K/Ks term and adsorption/desorption properties, but
also pore pressure. It varies with pore pressure instead
of remaining constant for the entire pressure range.
Based on Eq. (28), the third term is positive for any
pore pressure and the value of a may be larger than
unity.
Finally, we present a discussion about how the value of
a can be larger than unity for certain materials. Assuming
that the external pressure (P) and pore pressure (p) are the
same, the effective stress (Eq. 27) becomes:
Pe ¼ p� ap ¼ 1� að Þp ð34Þ
and the volumetric strain given by Eq. (26) becomes:
ev ¼1
K1� að Þp ð35Þ
When a\ 1, the volume of the material decreases with
increasing stress/pressure. This is the case for most porous
media, such as rocks; when a = 1, it is a single-phase solid
material and the effective stress law becomes meaningless;
when a[ 1, the volume of the material increases with
increasing stress/pressure. This is the case for self-
swelling/dilating materials, such as coal swelling with
adsorbing gas. The phenomenon (Harpalani and Chen
1995; Levine 1996; Robertson 2005; Liu and Harpalani
2013a) is well observed and documented in recent CBM
publications.
More generally, the effective stress tensor can be written
as:
reij ¼ rij � apdij and
a ¼ 1� KKsþ 3KaqRT lnð1þbpÞ
EAV0p
(ð36Þ
The proposed effective stress law has considered both
the shrinkage/swelling effect and porosity effect on the
stress–strain constitutive relationship for coal. Using this
effective stress concept, the complicated poro-elastic
problem for coal is reduced to that of a non-porous
material, and consequently solutions to static elastic
problems can be used for porous media problem with
both external stress and pore pressure variations. It has a
distinct advantage of simplicity and better representation of
the behavior of coal as a producing reservoir.
S. Liu, S. Harpalani
123
6 Summary and Conclusions
The concept of effective stress for porous media has been
addressed in several previous studies. These studies and the
resulting expressions for effective stress have been
reviewed in this paper. A new theoretical expression for the
effective stress law for deformation of coal is derived that
includes the shrinkage/swelling effect. It is based on the
newly proposed concept of shrinkage/swelling strain–stress
behavior. Apart from the theoretical modeling, experi-
mental work was carried out to check the validity of the
proposed stress–strain relationship. Based on the work
presented in this paper, the following conclusions are
made:
• The concept of shrinkage/swelling strain–stress is a
unique feature of coal reservoirs and it plays an
important role in the deformation of coal with contin-
ued gas production from coal reservoirs.
• A new effective stress relationship is derived. It
considers both the porosity effect and shrinkage/swell-
ing effect.
• When the compressibility of the bulk material is
sufficiently greater than that of the solid grain, and
there is no shrinkage/swelling effect, the effective stress
law, as first formulated by Terzaghi, is an excellent
approximation.
• Based on Eq. (28), the Biot coefficient (a) can have
values larger than 1 for self-swelling/dilation materials,
such as coal.
Overall, the proposed effective stress law provides a
new approach to analyze the elastic problem by reducing
three variables, namely the external stress, pore pressure
and shrinkage/swelling stress, down to one variable, termed
‘‘effective stress’’. Furthermore, to avoid confusion, it must
be emphasized that the effective stress law proposed in this
paper is only valid for solid elastic strain and does not hold
for inelastic processes. Also, only the unconstrained
hydrostatic condition was experimentally investigated in
this study.
Acknowledgments The authors thank Co-Editor Dr. Herbert Ein-
stein and two anonymous reviewers for valuable suggestions that
helped improve the manuscript.
References
Biot MA (1941) General theory of three-dimensional consolidation.
J Appl Phys 12:155–164
Biot MA (1955) Theory of elasticity and consolidation for a porous
anisotropic solid. J Appl Phys 26:182–185
Bishop AW (1955) The principle of effective stress. Tekniske
Ukeblad 39:859–863
Christensen NI, Wang HF (1985) The influence of pore pressure and
confining pressure on dynamic elastic properties of Berea
Sandstone. Geophysics 50:207–213
Detournay E, Cheng AH-D (1993) Fundamentals of poroelasticity. In:
Fairhurst C (ed) Comprehensive rock engineering: principles,
practice and projects, vol. II, analysis and design method.
Pergamon Press, New York, pp 113–171
Fatt I (1959) The Biot-Willis elastic coefficients for a sandstone.
J Appl Mech 26:296–297
Geertsma J (1957) The effect of fluid pressure decline on volumetric
changes of porous rocks. Trans AIME 210:331–340
George JG, Barakat MA (2001) The change in effective stress
associated with shrinkage from gas desorption in coal. Int J Coal
Geol 45:105–113
Harpalani S, Chen GL (1995) Estimation of changes in fracture
porosity of coal with gas emission. Fuel 74:1491–1498
Harpalani S, Mitra A (2010) Impact of CO2 injection on flow
behavior of coalbed methane reservoirs. Transp Porous Media
82:141–156
Hubbert MK, Rubey WW (1959a) Role of fluid pressure in mechanics
of overthrust faulting. Bull Geol Soc Am 70:115–205
Hubbert MK, Rubey WW (1959b) Role of fluid pressure in mechanics
of overthrust faulting: a reply. Bull Geol Soc Am 71:617–628
Izadi G, Wang S, Elsworth D, Liu J, Wu Y, Pone D (2011)
Permeability evolution of fluid-infiltrated coal containing dis-
crete fractures. Int J Coal Geol 85:202–211
Knaap WV (1959) Nonlinear behavior of elastic porous media. Trans
AIME 216:179–187
Lade PV, Boer RD (1997) The concept of effective stress for soil,
concrete and rock. Geotechnique 47(1):61–78
Levine JR (1996) Model study of the influence of matrix shrinkage on
absolute permeability of coalbed reservoirs. In: Gayer R, Harris I
(eds) Coalbed methane and coal geology. Geol Soc Special Pub,
London, pp 197–212
Li C (2000) The effective stress study of porous media and it
application. PhD dissertation, University of Science and Tech-
nology of China
Liu S, Harpalani S (2013a) A new theoretical approach to model
sorption-induced coal shrinkage or swelling. AAPG Bull
97(7):1033–1049
Liu S, Harpalani S (2013b) Permeability prediction of coalbed
methane reservoirs during primary depletion. Int J Coal Geol
113:1–10
Liu HH, Rutqvist J (2010) A new coal-permeability model: internal
swelling stress and fracture–matrix interaction. Transp Porous
Media 82:157–171
Ma Q, Harpalani S, Liu S (2011) A simplified permeability model for
coalbed methane reservoirs based on matchstick strain and
constant volume theory. Int J Coal Geol 85:43–48
Maggs FAP (1946) The adsorption-swelling of several carbonaceous
solids. Trans Faraday Soc 42:284–288
Moffat DH, Weale KE (1955) Sorption by coal of methane at high
pressures. Fuel 34:449–462
Nur A, Byerlee JD (1971) An exact effective stress law for elastic
deformation of rock with fluid. J Geophys Res 76:6414–6419
Nuth M, Laloui L (2008) Effective stress concept in unsaturated soils:
clarification and validation of a unified framework. Int J Numer
Anal Methods Geomech 32:771–801
Palmer I, Mansoori J (1998), How rermeability depends on stress and
pore pressure in coalbeds: a new model. SPE Reserv Eval Eng
539–544
Palmer I, Mavor M, Gunter B (2007) Permeability changes in coal
seams during production and injection. In: Proceedings of 2007
international Coalbed methane symposium. Paper 0713
Pan Z, Connell LD (2007) A theoretical model for gas adsorption-
induced coal swelling. Int J Coal Geol 69:243–252
Effective Stress Law for Deformation
123
Robertson EP (2005), Measurement and modeling of sorption-
induced strain and permeability changes in coal. Phd disserta-
tion, Colorado School of Mine, USA
Schiffman RL (1970) The stress components of a porous medium.
J Geophys Res 75:4035–4038
Seidle JP, Jeansonne DJ, Erickson DJ (1992) Application of
matchstick geometry to stress dependent permeability in coals.
In: SPE Rocky maintain regional meeting, Casper, Wyoming,
Paper SPE 24361, pp 433–444
Shi JQ, Durucan S (2004) Drawdown induced changes in permeabil-
ity of coalbeds: a new interpretation of the reservoir response to
primary recovery. Transp Porous Media 56:1–16
Skempton AW (1960) Effective stress in soils, concrete and rock. In:
Conference on pore pressure and suction in soils, Butterworths,
pp 4–16
Skempton AW, Bishop AW (1954) Soils, in building materials, their
elasticity and inelasticity. North Holland Publishing Company,
Amsterdam, pp 417–482
Suklje L (1969) Rheological aspects of soil mechanics. Wiley
Interscience, New York, p 123
Terzaghi K (1943) Theoretical soil mechanics. Wiley, New York
US EIA (2009) Natural gas gross withdrawals and production
released on 10/29/2010. US Energy Information Administration
Warpinski NR, Teufel LW (1992) Determination of the effective
stress law for permeability and determination in low-permeabil-
ity rocks. SPE Form Eval 7:123–131
Zhao Y (2009) Development and utilization of coal mine methane in
China. 2009. In: International Coalbed methane & Shale gas
symposium, Paper 0938. University of Alabama, Tuscaloosa,
pp 65–74
Zhao Y, Hu Y, Wei J, Yang D (2003) The experimental approach to
effective stress law of coal mass by effect of methane. Transp
Porous Media 53:235–244
Zimmerman RW, Somerton WH, King MS (1986) Compressibility of
porous rocks. J Geophys Res 91:765–777
S. Liu, S. Harpalani
123
