Advances in Oil and Gas Exploration & Production
Vladimir Karev
Yuri Kovalenko
Konstantin Ustinov
Geomechanics
of Oil and Gas
Wells
Advances in Oil and Gas Exploration &
Production
Series Editor
Rudy Swennen, Department of Earth and Environmental Sciences,
K.U. Leuven, Heverlee, Belgium
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Vladimir Karev Yuri Kovalenko
Konstantin Ustinov
•
•
Geomechanics of Oil
and Gas Wells
123
Vladimir Karev
Ishlinsky Institute for Problems in
Mechanics
Russian Academy of Sciences
Moscow, Russia
Yuri Kovalenko
Ishlinsky Institute for Problems in
Mechanics
Russian Academy of Sciences
Moscow, Russia
Konstantin Ustinov
Ishlinsky Institute for Problems in
Mechanics
Russian Academy of Sciences
Moscow, Russia
ISSN 2509-372X
ISSN 2509-3738 (electronic)
Advances in Oil and Gas Exploration & Production
ISBN 978-3-030-26607-3
ISBN 978-3-030-26608-0 (eBook)
https://doi.org/10.1007/978-3-030-26608-0
© Springer Nature Switzerland AG 2020
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Preface
This book presents an integrated approach to the study of geomechanical
processes occurring in oil- and gas-bearing formations during their development. It includes: the choice of a model that takes into account the basic
properties of rocks; experimental finding of model parameters; and numerical
modeling as well as direct physical modeling of deformation and filtration
processes in reservoir and host rocks. The main features of the behavior of
rocks are taken into account, such as anisotropy of the mechanical properties
of rocks during elastoplastic deformation, dependence of permeability on
the total stress tensor, contribution of the filtration flow to the stress state
of the formation, and influence of not only tangential but also normal stresses
on the transition to inelastic deformation. It is shown how the presented
approach allows solving practical problems of increasing the productivity of
wells, oil recovery, and ensuring the stability of wellbores.
This book is intended for specialists-geomechanics working primarily
in the oil and gas sector, teachers, graduate students, and students of oil
universities and faculty, as well as for all those who are interested in scientific
and technological development and meeting the enormous needs of mankind
in raw materials and energy.
Moscow, Russia
Vladimir Karev
Yuri Kovalenko
Konstantin Ustinov
v
Introduction
Oil and gas are currently the most important geological resources on our
planet. The importance of oil and gas is not limited to their dominant role in
the fuel supply to the national economy. These resources are also the most
valuable and indispensable industrial and strategic raw materials for the
production of many different motor fuels, oils and lubricants, road surfaces,
paraffin, and petrochemical products.
Oil is produced in 80 countries around the world. Oil and gas play a
crucial role in the development of any country’s economy. Natural gas is
very convenient for pipeline transportation and combustion, cheap energy,
and household fuel. All types of liquid fuels are produced from oil: gasoline,
kerosene, jet and diesel fuel, gas turbine fuel for locomotives, and fuel oil for
boiler units. High-boiling fractions of oil are used to produce a huge range of
lubricants, especially oils and greases. Oil is also used to produce paraffin,
carbon black for the rubber industry, petroleum coke, numerous bitumen
grades for road construction, and many other commercial products.
Modern oil and gas production technologies are largely based on the
drilling of inclined and horizontal wells. However, there are serious problems
with their use. It turned out that the stability of inclined wellbores
significantly depends on the strain and strength characteristics of rocks, the
presence and degree of their anisotropy, as well as the geometry of the wells
and the pressure on their bottom hole. As a rule, complications when drilling
oil and gas wells related to the loss of wellbore stability are accompanied by
large expenditures for the elimination of their consequences. Therefore, the
forecasting and prevention of this type of complications play an important
role in reducing the cost of well construction. In addition, wellbore
destruction is one of the main factors limiting the maximum flow rates of
wells.
In recent years, physical modeling and mathematical modeling of
geomechanical processes in oil and gas reservoirs have become increasingly
important in global and Russian practices. This is primarily due to the
increasing complexity of the well profile, the increase in the length of
horizontal boreholes, the use of complex drilling techniques such as
underbalanced drilling, as well as with the increasingly complex geological
conditions of drilling and operating wells.
With rising costs, especially when implementing projects in harsh climatic
conditions, in hard-to-reach regions or at sea, pre-drilling modeling, the
so-called drilling on paper, becomes an important element of well
vii
viii
construction planning, as it helps to minimize costs, reduce nonproductive
time, and improve drilling efficiency.
With the help of geomechanical modeling, it is possible to assess the
behavior and changes in the environment during drilling and field
development, to predict the pore pressure, to assess the properties of
reservoir formation, to determine the values of stress in formations, to assess
the stability of the walls of the well, to calculate the optimal trajectory of the
wellbore, and to optimize the process of drilling the well.
Currently, a number of fields developed by oil companies are characterized by a significant degree of fracturing of the reservoir rocks. Fractured
reservoirs have a number of features, including a complex dependence of
filtration properties of the rock on the local stress–strain state. Availability of
fracturing causes substantially anisotropic character of this dependence. The
permeability of a crack depends not only on the pressure of liquid in it and
the first invariant of the stress tensor, but also on the difference between the
stresses acting in the crack plane and normal stresses. As it was discovered in
the course of core study experiments, the effect of stresses on rock
permeability cannot be neglected; this leads to the need for a detailed analysis
of the stress–strain state and its dynamics during field development. It should
be noted that the change in permeability due to changes in the stress state
affects the distribution of pore pressure in the reservoir which in turn affects
the redistribution of effective stresses. Thus, there is a task to organize the
process of modeling which takes into account the mutual influence of
hydrodynamic and geomechanical processes.
The traditional approach to solving such problems is to create mechanical
and mathematical models and find answers to these questions with their
help. However, attempts to create an adequate mechanical and mathematical
model describing the processes of deformation and destruction in the vicinity
of an inclined well in rock with pronounced anisotropy of elastic and strength
properties lead to its significant complication.
In turn, the complexity of the model inevitably leads to an increase in the
number of deformation and strength parameters included in the model.
Experimental determination of these parameters for anisotropic rocks is in
itself a complex problem requiring sophisticated laboratory techniques and
equipment. In addition, any mathematical model requires the adoption of
some strength law, which is also a separate challenge for anisotropic rocks.
All of this leads to the need for certain simplifications and assumptions in
the model, resulting in practical conclusions based on the calculations by use
such models which are often only evaluative in nature.
The approach presented in the monograph differs radically from that
described above. It is based on the direct physical modeling of rock
deformation and destruction processes in the vicinity of a well on a true
triaxial test facility under the real stresses arising near various geometries of
wells and at various bottom-hole pressures. The loading program of the
researched samples is determined on the basis of the mechanical and
mathematical models taking into account anisotropy of deformation and
strength properties of rocks.
Introduction
Introduction
ix
This book presents the results of geomechanical studies of deformation
and destruction processes in the bottom-hole formation zone, carried out with
the purpose of predictive risk assessment of its uncontrollable destruction and
the development of measures to prevent them, as well as to provide
enhancing productivity of well based on the control of the stress–strain state.
The comprehensive studies conducted include experimental studies of
structural–lithological, strain–strength and filtration characteristics of rocks
of fields, mathematical modeling and physical modeling of deformation and
destruction processes occurring in a rock massif during the development of a
field.
This book is intended for specialists of the oil and gas sector, teachers,
postgraduates, and students of oil universities and faculty, as well as for all
those who are interested in issues of scientific and technological development
and meeting the enormous needs of mankind in raw materials and energy.
Contents
1
Stress-Strain State of Rocks . . . . . . . . . .
1.1 Elastic Deformation . . . . . . . . . . . .
1.2 Transition to Inelastic Deformation .
References . . . . . . . . . . . . . . . . . . . . . . . . .
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1
5
22
2
Deformation and Fracture of Rocks in the Presence
of Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Filtration in Reservoir . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Equations of Poroelasticisity . . . . . . . . . . . . . . . . . . .
2.3 Inelastic Deformation with Regard to Filtration . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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25
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Mechanical and Mathematical, and Experimental
Modeling of Oil and Gas Well Stability . . . . . . . . . . . . . . . . . .
3.1 Stress State in the Vicinity of the Well in Isotropic
Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Mechanical Model of Stability of Inclined and
Horizontal Wells in Anisotropic (Layered) Formations . . .
3.3 Stress State in the Vicinity of the Well in Elastically
Anisotropic Rocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Physical Simulation of Conditions in the Vicinity of
Inclined and Horizontal Wells in Anisotropic (Layered)
Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Equipment for Studying Deformation and Strength
Properties of Rocks in Triaxial Loading . . . . . . . . . . . . .
4.1 Karman Type Installations . . . . . . . . . . . . . . . . . . . .
4.2 True Triaxial Loading Systems . . . . . . . . . . . . . . . . .
4.3 Examples of True Three-Axis Loading Installations .
4.4 Triaxial Independent Loading Test System TILTS . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Loading Programs for Rock Specimens on Triaxial
Independent Loading Test System (TILTS) . . . . . . . . . . . . . . .
5.1 Determining Strength and Elastic Characteristics
of Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
36
45
50
59
59
61
61
64
65
67
70
71
71
xi
xii
Contents
5.2
Programs for Physical Modeling of Deformation
Processes in the Vicinity of Inclined and Horizontal
Wells in Isotropic and Anisotropic (Layered)
Formations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Hollow Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
79
82
6
Dependence of Permeability on Stress State . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
96
7
Influence of Filtration on Stress–Strain State and Rock
Fracture in the Well Vicinity . . . . . . . . . . . . . . . . . . . . . . . . . . 97
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8
Results of Tests of Rock Specimens by Using TILTS . . .
8.1 Results of Physical Modeling of Resistance
to Failure of Inclined and Horizontal Wells for
Particular Objects . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Determination of Parameters of Models of Plastic
Deformation for Transverse Isotropic Reservoir and
Host Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
Mathematical Modeling of Mechanical and Filtration
Processes in Near-Wellbore Zone . . . . . . . . . . . . . . . . . . .
9.1 Calculation of the Inelastic Deformation Zone in the
Absence of Filtration. . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Calculation of Zone of Inelastic Deformation
in Case of Filtration; The Algorithm . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
....
107
....
107
....
....
118
139
....
141
....
141
....
....
145
152
10 Directional Unloading Method is a New Approach to
Enhancing Oil and Gas Well Productivity . . . . . . . . . . . . . . . .
10.1 Technology of Directional Unloading a Reservoir . . . . . .
10.2 Methodology for Well Productivity Enhancing by
Means of Directional Unloading . . . . . . . . . . . . . . . . . . . .
10.3 Practical Implementation of the Directional Unloading
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
155
155
161
164
166
Notations
Að k Þ
A; B
BCð0Þ
Yield strength;
Material constants in Drucker–Prager’s
criteria;
Material constants in Lui–Huang–Stout
plasticity criteria;
Material constants in Lui–Huang–Stout
plasticity potential;
Material constants in Caddell–Raghava–
Atkins plasticity criteria;
Material constants in Caddell–Raghava–
Atkins potential;
Average value of parameters BCðiÞ of mod-
BLð0Þ
ified Hill’s plasticity model in the form of
Caddell–Raghava–Atkins;
Average value of parameters BCðiÞ of mod-
BLð1Þ ; BLð2Þ ; BLð3Þ
LQ
LQ
BLQ
ð1Þ ; Bð2Þ ; Bð3Þ
BCð1Þ ; BCð2Þ ; BCð3Þ
CQ
CQ
BCQ
ð1Þ ; Bð2Þ ; Bð3Þ
C1 ; C2 ; C3 . . .
C11 ; C12 ; C13 ; C33 ; C44 ; C66
E
E1
E3
Ep
G12
G13
F
FL
ified Hill’s plasticity model in the form of
Lui–Huang–Stout;
Constants of integration;
Elastic constants of transverse isotropic
medium;
Young’s modulus;
Young’s modulus of transverse isotropic
medium in the plane of isotropy;
Young’s modulus of transverse isotropic
medium along the normal to the plane of
isotropy;
Modulus of plasticity;
Shear modulus of transverse isotropic
medium in the plane of isotropy;
Shear modulus of transverse isotropic
medium in planes normal to the plane of
isotropy;
Yield criterion; criterion of transition to
inelastic state;
Modified Hill’s yield criterion in the Lui–
Huang–Stout form;
xiii
xiv
FC
H
H
GH
ð11Þ ; Gð22Þ ; Gð33Þ
GLð11Þ ; GLð22Þ ; GLð33Þ
GCð11Þ ; GCð22Þ ; GCð33Þ
H
I1
I2
K
H
H
LH
ð12Þ ; Lð13Þ ; Lð23Þ
LLð12Þ ; LLð13Þ ; LLð23Þ
LCð12Þ ; LCð13Þ ; LCð23Þ
Q
QL
QC
T
V
W
a11 ; a12 ; a13 ; a33 ; a44 ; a66
dk
dk
fj
h
ga
m
n
ni
pi
p
q
Notations
Modified Hill’s yield criterion in the
Caddell–Raghava–Atkins form;
Material constants in Hill’s plasticity
criterion;
Material constants in modified Hill’s plasticity criterion and plasticity potential in
the form of Lui–Huang–Stout;
Material constants in modified Hill’s plasticity criterion and plasticity potential in
the form of Caddell–Raghava–Atkins;
Material function, the inverse of plasticity
modulus Ep ;
First invariant of stress tensor;
Second invariant of stress deviator r0jk ;
Modulus of compression;
Material constants in Hill’s plasticity
criterion;
Material constants in modified Hill’s
plasticity criterion and plasticity potential
in the form of Lui–Huang–Stout;
Material constants in modified Hill’s plasticity criterion and plasticity potential in
the form of Caddell–Raghava–Atkins;
Plasticity potential;
Plasticity potential of modified Hill’s
model in the form of Lui–Huang–Stout;
Plasticity potential of modified Hill’s
model in the form of a Caddell–Raghava–
Atkins;
Temperature;
Relative change in the volume of pores;
Elastic energy;
Compliances of transversally isotropic
medium;
Parameter of isotropic hardening;
Coefficient of proportionality between
increase in plastic strains and derivative
of plasticity potential
Density of volume forces;
Depth;
Function in the law of kinematic
hardening;
Rock porosity;
Normal vector;
Direction cosines in laboratory coordinate
system;
Components of traction;
Pore pressure;
Rock pressure;
Notations
xv
qf
qf
r; /; z
sij
si
s1 ; s2 ; s3
u
ui
x1 ; x2 ; x3
D
Kijkl
aDP
aT
aP
aij
c
d
dij
e
eij
eTij
eEij
ePij
g
h
j
jij
k
l
l0 ; k0
m
m12
m13 ; m31
q
q0
qc
r
rij
r1 ; r2 ; r3
Density of fluid flow;
Radial component of density of fluid flow;
Cylindrical coordinates;
Components of tensor of effective stress;
Intensity of effective shear stresses;
Principle effective stresses;
Displacement;
Components of displacement vector;
Cartesian coordinates;
Laplace operator;
Components of tensor of elasticity;
Constant in an alternative form of the
Drucker–Prager criterion;
Volumetric thermal expansion;
Biot’s coefficient;
Parameters of kinematic hardening;
Specific weight;
Ratio of contact areas of the skeleton
grains and gross area of cross section;
Kronecker’s unit tensor;
Strain;
Components of strain tensor;
Components of total strain tensor;
Components of elastic strain tensor;
Components of plastic strain tensor;
Dynamic viscosity;
An angle between a well axis and the
vertical;
Permeability;
Components of permeability tensor;
Lamé’s first constant;
Lamé’s second constant (shear modulus);
Lamé’s constants in poroelasticity;
Poisson’s ratio;
Poisson’s ratio of transverse isotropic
medium in the plane of isotropy;
Poisson’s ratios of transverse isotropic
medium in a plane normal to the plane of
isotropy;
Fluid density;
An angle of internal friction in the
Mohr–Coulomb criterion;
An angle of friction in planes of weakening;
Stress;
Components of stress tensor;
Principle stresses;
xvi
ri
rm
rn
r0jk
rY
rY 11 ; rY 22 ; rY 33
rY 12 ; rY 23 ; rY 13
rC1 ; rC2 ; rC3
rS12 ; rS23 ; rS13
rT1 ; rT2 ; rT3
s
s0
sc
ss
u
Notations
Stress intensity;
Hydrostatic stress;
Normal stress;
Components of deviator of stress;
Yield stress;
Yield stresses of anisotropic medium along
corresponding directions;
Yield shear stresses of anisotropic medium
in corresponding planes;
Compressive yield stresses of anisotropic
medium along the principle directions;
Shear yield stresses of anisotropic medium
in corresponding planes;
Tensile yield stresses of anisotropic medium along the principle directions;
Shear stress;
Cohesion in the Mohr–Coulomb criterion;
Adhesion in the Mohr–Coulomb criterion
for planes of weakening;
Constant in alternative form of the
Drucker–Prager criterion;
The angle between the radius vectors
of the point on the contour of the well
and the vertical; the angle between the
maximum compressive stress and the
formation plane
Vector values are written in bold, and vector components are written in italic
with index. Summation is assumed for repeated tensor indices (indices for
which summation is not supposed to be taken in brackets); after-comma
indices in tensor values denote a derivative over the corresponding
coordinate. Tensile stresses and strains are considered positive.
1
Stress-Strain State of Rocks
By definition, geomechanics is a theoretical and
applied science of the mechanical properties of
rocks and mechanical processes, which studies
the stress-strain fields that occur in a specific
physical environment (Baklashov and Kartozia
1975). In modern petroleum industry, geomechanics is a discipline that combines rock
mechanics, geophysics, petrophysics and geology in order to calculate the Earth’s response to
any changes in rock stress, pore pressure and
temperature of the reservoir and host rocks.
Geomechanical modeling includes experimental, analytical, and numerical methods.
Experimental models are based on data of physical
and mechanical laboratory tests conducted on rock
specimens. Such tests provide valuable information about the properties of the rock, but they are
quite expensive and time-consuming.
Geomechanical modeling is based on continuum mechanics, namely: the theory of elasticity,
the theory of plasticity, fracture mechanics, the
theory of filtration.
The classical theory of elasticity is based on a
perfectly elastic model of a deformable solid
body. Such a body is characterized by the simple
linear relationship between stress and strain.
1.1
Elastic Deformation
Mechanical behavior of rocks is rather specific
comparing to other solids; this peculiarity is
related to the presence of a wide diversity of
© Springer Nature Switzerland AG 2020
V. Karev et al., Geomechanics of Oil and Gas Wells,
Advances in Oil and Gas Exploration & Production,
https://doi.org/10.1007/978-3-030-26608-0_1
structures, such as graininess, fracturing, geological disturbances (Goodman 1980; Jaeger
2007; Baklashov and Kartozia 1975), so that for
any allocated elementary volume there will
always be elements of structure of the comparable scale. This makes application of the mathematical apparatus of solid-state mechanics
questionable, since the basic concept of elementary volume cease to be rigorous, but due to
the lack of alternative, the theories of elasticity,
plasticity and filtration serve nevertheless as the
basis for describing the processes in rock masses.
Another peculiarity of rock deformation consists
in accumulation of irreversible deformations
even under relatively small stresses, which
restricts the accuracy of calculations using traditional plasticity theories. In addition, the
inelastic behavior of rocks differs from the plasticity of metals by its physical nature- except for
clays, inelasticity is more often associated with
the accumulation of micro-injuries, the average
macroscopic contribution of which, however,
appeared similar to the manifestation of the
classic plasticity of metals. Therefore, term
“inelasticity” will be preferably used hereafter,
and the term plasticity can be taken in quotation
marks.
The presence of structure (at a thin level) and
texture (at a coarser level) in rocks leads to the
appearance of anisotropy of physical and
mechanical properties. Among the most significant texture elements resulting to appearance of
anisotropy, we should mention layering and
1
2
1 Stress-Strain State of Rocks
fracturing. The layering inherent to almost all
sedimentary and metamorphic, and sometimes
magmatic effusive rocks, causes the presence of a
distinguish direction (perpendicular layering)
and, thus, the appearance of a transverse isotropy
of properties described by tensor values. The
presence of additional texture elements not related to layering, such as systems of directional
cracks, leads to complication of the picture and
reduction of the type of symmetry to lower
levels.
In case of transverse isotropy of the properties
described by second-rank tensors (thermal conductivity, filtration, thermal expansion, etc.), the
components of tensors are expressed in terms of
two independent constants (characterizing the
corresponding properties in the layering plane
and along its normal); the components of the
elastic tensor (fourth rank) are expressed in terms
of five independent constants.
The mechanical state is characterized by
kinematic and force values. Among the former
are displacement vector and strain tensor, among
the latter is the stress tensor. Within the framework of the theory of small deformations, only
which will be considered hereafter, the complete
strain eTij can be divided into elastic eEij and
non-elastic (plastic) ePij components
eTij ¼
1
ui;j þ uj;i ¼ eEij þ ePij
2
ð1:1Þ
Note that, in general, only complete strains are
assumed to be subject to the conditions of compatibility, rather than elastic and inelastic parts
separately. This assumption is equivalent to the
one of existence of an initial state with zero both
elastic and inelastic strains, starting with this
state first inelastic deformation was imposed
following elastic deformation occurred compensating incompatibility. Thus, it is the total strain
that is bound to the displacement vector ui by the
Cauchy relations
1
ð1:2Þ
eTij ¼ ui;j þ uj;i
2
Equations (1.1) and (1.2) determine, therefore, all kinematic description of the
environment.
The components of the stress tensor rij are
related to each other by equilibrium equations
rij;i þ fj ¼ 0
ð1:3Þ
Here fj —is the density of volumetric forces,
which are most often gravity forces. In most
practical cases, when dealing with rocks lying at
the depths of several kilometers, the change of
forces caused by gravity within the area of
interest can be neglected (Goodman 1980; Jaeger
2007; Baklashov and Kartozia 1975), so often
the problems are solved in the formulation of the
absence of volumetric forces.
rij;i ¼ 0
ð1:4Þ
Equation (1.3) define a static description. For
dynamic problems, inertia terms should be added
according to d’Alembertprinciple.
To close the system, it is necessary to determine the relationship between stresses and
strains. The type of the relationship can vary
significantly for various media and is determined
by the physical characteristics of the medium; the
relationship can include functional dependencies
between stresses, strains, their derivatives and
other variables. In the absence of inelastic
deformations in the simplest case, the relationship between stress and deformation is determined by the generalized Hooke’s law. This
relationship is assumed to be preserved (for the
elastic component of the strain tensor eEij ) in the
presence of inelastic part of deformation (e.g., De
Wit 1970, 1973). In case of anisotropic medium
characterized by a fourth-rank elastic tensor Kijkl ,
Hooke’s law has the form:
rij ¼ Kijkl eElk
ð1:5Þ
In the most general case, the components of
the elasticity tensor are expressed in terms of 21
independent constants. However, such a
description is necessary only for single crystals
1.1 Elastic Deformation
3
that have a certain, very low degree of internal
symmetry. In case of isotropy, the situation is
simplified:
rij ¼ kdij eEkk þ 2leEij
ð1:6Þ
Here k; l are Lamé constants; l makes sense
as shear module.
In some cases, it is more convenient to write
Hooke’s law in form of strain dependence on
stresses. For isotropy
1
m
m
r11 r22 r33
E
E
E
m
1
m
¼ r11 þ r22 r33
E
E
E
m
m
1
¼ r11 r22 þ r33
E
E
E
eE33
eE12 ¼
1
r12 ;
2l
eE13 ¼
ð1:7Þ
1
1
r13 ; eE23 ¼
r23
2l
2l
Here E; m are Young’s module and Poisson’s
ratio. In linear elasticity theory, there are only
two independent constants characterizing isotropic bodies (e.g., Papkovich 1939; Rabotnov
1988; Landau and Lifshits 1987). This follows
from the existence of the elastic potential, which
in linear theory is a square form of strains, and in
isotropic medium is a function of strain tensor
invariants; since it is possible to construct exactly
two quadratic invariants from the components of
the tensor of strain, the symmetric tensor of the
second rank (for example, the square of the first
invariant and the second invariant), the most
general form for the elastic energy is
W¼
k E 2
e
þ leEij eEij
2 kk
ð1:8Þ
Thus, appearance of Lamé constants follows
most naturally from representation (1.8). Formulas of Hooke’s law (1.6) immediately follow
(1.8) in accordance with Lagrange theorem. The
relationship between Lamé constants and technical elastic constants E; m looks like
E
mE
;k ¼
2ð 1 þ m Þ
ð1 þ mÞð1 2mÞ
ð1:9Þ
For transverse isotropy (Lekhnitsky 1950,
1977), which is typical for most sedimentary
rocks
1
m12
m13
r11 r22 r33
E1
E1
E1
m12
1
m13
¼
r11 þ
r22 r33
E1
E1
E1
m31
m31
1
¼
r11 r22 þ
r33
E3
E3
E3
1
1
1
¼
r12 ; eE13 ¼
r13 ; eE23 ¼
r23
2G12
2G13
2G13
ð1:10Þ
eE11 ¼
eE22
eE11
eE11 ¼
eE22
l¼
eE12
Here axis x3 is normal to the isotropy plane.
Suchaconventionisacceptedinmechanicsofsolidstateandwillbeusedhereafterwhile describing rock
properties and mathematical modeling; an alternative numbering will be used for describing
experiments (Chaps. 6–8). For rocks, the isotropy plane coincides with the bedding planes;
for quite frequent horizontal occurrence, the axis
x3 coincides with the vertical. Among the constants introduced in the formulas (1.10), only five
are independent: modulus of elasticity in the
isotropy plane and in the planes normal to it,
E1 ; E3 , Poisson’s ratios in the plane of isotropy.
m12 , one of the two Poisson’s ratios in the plane
normal to the plane of isotropy, for example, m31 ,
shear modulus in the plane normal to the isotropy
plane, G13 . Shear module G12 in the plane of
isotropy is related to the modulus of elasticity
and Poisson’s ratio in this plane as usually
1
2ð1 þ m12 Þ
¼
G12
E1
Another relation
reciprocity theorem
follows
m13 m31
¼
E1
E3
ð1:11Þ
from
Betty’s
ð1:12Þ
4
1 Stress-Strain State of Rocks
With the use of the last ratio the Eq. (1.10) are
usually written in the following form (Lekhnitsky 1950, 1977)
1
m12
m31
r11 r22 r33
E1
E1
E3
m12
1
m31
¼
r11 þ
r22 r33
E1
E1
E3
m31
m31
1
¼
r11 r22 þ
r33
E3
E3
E3
1
1
1
¼
r12 ; eE13 ¼
r13 ; eE23 ¼
r23
2G12
2G13
2G13
ð1:13Þ
eE11 ¼
eE22
eE11
eE12
The relationship between the compliance and
elastic coefficients and technical constants
(Young’s modules, shear and Poisson’s coefficients) can be obtained by resolving the system
(1.15) with respect to strains and comparing the
stress coefficients with those in (1.13)
1
¼ G13 ¼ C44
a44
1
E1
¼ C66
¼ G12 ¼
a12
2ð1 þ m12 Þ
2
1
C11 C33 C13
a11 ¼
¼
2
E1 ðC11 C12 Þ ðC11 þ C12 ÞC33 2C13
2
m12
C12 C33 C13
¼
2
E1 ðC11 C12 Þ ðC11 þ C12 ÞC33 2C13
m13 m31
C13
¼
¼
¼
2
E1
E3 ðC11 þ C12 ÞC33 2C13
1
C11 þ C12
¼
¼
2
E3 ðC11 þ C12 ÞC33 2C13
a12 ¼
Note that the presence of two Poisson’s ratios
in the plane normal to the plane of isotropy,
m31 ; m13 , even if one of them is only implied
rather than used, in the above representations
causes ambiguity and is a potential source of
errors. Although a similar form, developed in the
works of Chentsov (1936), Rabinovich (1946),
Sekerzh-Zenkovich (1931), has now become
common and is used in the main industrial
packages (ANSYS, SOLID), a form of recording
in terms of the matrix of coefficients of compliance (Lekhnitsky 1950, 1977) that is freeform
potential sources of error seems preferable
eE11 ¼ a11 r11 þ a12 r22 þ a13 r33
eE22 ¼ a12 r11 þ a12 r22 þ a13 r33
eE33 ¼ a13 r11 þ a13 r22 þ a33 r33
2eE12 ¼ a66 r12 ; 2eE13 ¼ a44 r13 ; 2eE23 ¼ a44 r23
ð1:14Þ
Similarly, the relation may be written in terms
of the inverse matrix (matrix of elastic
coefficients)
r11 ¼ C11 eE11 þ C12 eE22 þ C13 eE33
r12 ¼ 2C66 eE12 ;
r13 ¼ 2C44 eE13 ;
r23 ¼ 2C44 eE23
a33
ð1:16Þ
or by resolving the system of Eq. (1.13) with
respect to stresses and comparing the strain
coefficients with those of (1.15).
Special attention should be paid to the problem of plane strain, when displacements along
one selected axis are assumed to be absent. Such
states are typical for wells and long excavations.
If the selected axis coincides with the normal to
the plane of elastic symmetry, the stress distribution can be considered as for an isotropic
body. In general case of an arbitrary angle
between these axes the problem is usually solved
numerically 3-D and no additional consideration
is required. The case when the axis of absence of
displacements lies in the plane of isotropy is of a
particular interest. Equations of Hooke’s law can
be written down as follows (Lekhnitsky 1950,
1977)
eE11 ¼ b11 r11 þ b13 r33
r22 ¼ C12 eE11 þ C12 eE22 þ C13 eE33
r33 ¼ C13 eE11 þ C13 eE22 þ C33 eE33
a13
ð1:15Þ
eE33 ¼ b13 r11 þ b33 r33
2eE13
ð1:17Þ
¼ b55 r13
Here constants b make sense as constants
modified for plane strain:
1.1 Elastic Deformation
a212
a11
a2
¼ a33 13
a11
a13 a12
¼ a13 a11
¼ a55
5
1.2
b11 ¼ a11 b33
b13
b55
ð1:18Þ
Similar to the isotropic case, the difference
between constants characterizing plane strain and
plane stress is usually not drastic for practical
cases.
Therefore, the closed system of equations of
elasticity consists in three groups of equations:
equations of equilibrium (1.3) for stresses, kinematics Eq. (1.1) for strains and displacements,
constitutive equations (Hooke’s law) in one of
the form of (1.5)–(1.7), (1.13)–(1.15) relating
stress and strain tensors.
Substitution of kinematics Eq. (1.1) into
Hooke’s law (1.6) and then substitution of the
result into equations of equilibrium (1.3) leads to
three scalar equations with respect to three
components of displacement vector known as
Lamé equations
ðk þ lÞui;ij þ luj;ii þ fj ¼ 0
ð1:19Þ
In polar coordinates for axi-symmetrical
problem equations of equilibrium and Lamé
equation have the form (e.g. Tymoshenko and
Goodyear 1979)
@rr
rr r/
þ
þ fr ¼ 0
@r
r
@ 1@
ðk þ 2lÞ
ðrur Þ þ fr ¼ 0
@r r @r
ð1:20Þ
ð1:21Þ
Here ur is radial displacement; rr ; r/ are
radial and circumferential stresses; one index
instead of two repeated indexes will be used
sometimes throughout the text for normal components of stress and strain tensors.
Transition to Inelastic
Deformation
The growing complexity of field processing
conditions requires the development and
improvement of well drilling and hydrocarbon
production technologies. Modern technologies of
the oil and gas field development using directional and horizontal wells require accounting for
the anisotropy of deformation and strength
properties of rocks composing productive and
host formations. The issues of wellbore stability
during drilling and processing, the choice of
optimal modes of technological operations
inevitably lead to the need for preliminary
geomechanical modeling of the processes of
deformation and destruction of rocks under various technological impacts.
On Model Choice
At present, a large number of models reflecting the
variety of observed properties have been created
and successfully used to describe rock deformation and destruction. However, the interest of the
developers of many of the existing models has
been focused on certain particular aspects of the
problem, while ignoring other aspects. It is especially unfortunate when such distortions are
determined by the current externally imposed
“trends” rather than by attempts to identify the
most significant details of rock behavior.
Below is the approach to modeling is described, which according to the authors reflects the
key features of rock behavior, namely:
– influence of not only tangential but also normal
stresses on transition to inelastic deformation;
– anisotropy of elastic and strength properties;
– possible presence of volumetric inelastic
strain and its nontrivial dependence on the
stress state: at least absence of proportionality
of inelastic volumetric strain to the current
volumetric stress.
6
A sufficient number of models have been
proposed to describe the contribution of each of
these features to the deformation process at different times. Thus, the influence of normal
stresses on the transition to an inelastic state,
peculiar to rocks, is accounted for in the criteria
of Coulomb (1776), Drucker and Prager (1952),
Barton (1971), Barton (1976) Goodman (1980),
Barton (2011). All these criteria were formulated
for isotropic media. However, the majority of
rocks reveal anisotropy of not only elastic properties, but also properties describing the
elastic-plastic transition and properties describing
plastic deformation.
To describe the plastic deformation of anisotropic media, several variants of the plasticity
theory have been proposed (Hill 1948; Novozhilov 1963; Lomakin 1980, 1991, 2000, Chanyshev 1984; Myasnikov and Oleynikov 1984;
Annin 2011, 2016), both in terms of deformation
theory (Lomakin 1991) and the theory of plastic
flow (Chanyshev 1984; Lomakin 2000). Among
the trends of the theory of anisotropic plasticity,
one can distinguish the approach based on
decomposition of the stress fields in tensor bases
(Novozhilov 1963; Chanyshev 1984 and the
approach based on the description of the yield
surface in the space of stresses in the coordinate
frame associated with the axes of isotropy of the
material by a quadratic form of the general type
(Hill 1948). Both approaches have been developed: the first one mostly for construction analytical and semi-analytical solutions (Chanyshev;
Imamutdinov and Chanyshev 1988), the second
one for the finite element modeling. Various
variants of the theory of plasticity are successfully used to solve the problems of mountain
mechanics; among the recent we note (Kurlenya
et al. 2014; Salganik et al. 2015; Protosenya and
Karasev 2016). Among the existing criteria of the
transition to inelastic deformation, we will
highlight those that take into account both anisotropy and the influence of normal stresses
(Caddel et al. 1973; Deshpande et al. 2001).
1 Stress-Strain State of Rocks
To describe deformation accompanied by
inelastic changes in volume, the concept of dilatancy was introduced by Reynolds (1885). The
concept was developed in works (Mead 1925;
Nikolaevsky 1967, 1996). According to this
concept the tensor of plastic strains is decomposed
into deviatoric and volumetric parts, and for the
first of them, as a rule, constitutive laws traditional
for plasticity (most often associated flow law) are
applied, and for the second one, an additional
constitutive law, usually of empirical nature, is
written. Note that such a description implies the
violation of the associated flow law: associativity
is maintained for the deviatoric part of plastic
deformations only. Usage of associativity for
complete plastic strains, including the volumetric
part, would lead to unrealistic results in the
description of mechanical behavior. In particular,
according to the fully associated law, a large
inelastic volume change, comparable to the
intensity of plastic strains, should have been
observed during rock compression.
The mentioned models, describing separately
the main noted features of rock deformation,
form the basis for solving problems of geomechanics. Moreover, on the basis of many of these
models program codes implemented into the
modern calculation systems are created. At the
same time, there are a number of difficulties in
directly using these models for solving problems
of geomechanics—each of the models, taking
into account one or several features of the
mechanical behavior of rocks, does not take into
account others.
Thus, one of the main tasks is the development and adaptation of the inelastic behavior
models for the problems of geomechanics
accounting listed major features of rocks.
Traditional Yield Criteria
Under growing stresses, on reaching some critical stress magnitude, solids cease to deform in a
purely elastic way. Various types of failure are
observed: from purely brittle fracture, when the
1.2 Transition to Inelastic Deformation
7
global loss of strength occurs almost instantly
without intermediate inelastic deformation, to
multistage degradation, when transition to
inelastic state precedes global failure, and
inelastic components of strains are accumulated
during significantly period while carrying
capacity of the material is still preserved. Unlike
traditional structural materials such as steel, the
transition to inelasticity of rocks may not be well
pronounced. Moreover, inelasticity is in some
cases manifested even at very low stresses, from
the beginning of loading. Therefore, the criteria
of inelastic transition may be somewhat arbitrary.
The situation is aggravated by the fact that for
rocks due to their great variety and the lack of
well-developed database as for steels, there are
no standardized criteria such as the criteria for
metals and alloys, where the technical yield
strength is taken as the value of stress, at which
the residual strain reaches a particular magnitude
of 0.2% or 0.5%.
The wide difference in the response of rocks
to mechanical impact makes the problem of
mathematical description of strength criteria not
obvious. The most traditional strength criteria are
those of Mohr-Coulomb (1776) and Drucker and
Prager (1952). A very detailed description of
these criteria is contained in monograph by
Nadai (1950).
to note that Henri Tresca formulated the plasticity criterion as a generalization of experimental
data 90 years after the publication of Charles
Augustin de Coulomb. Therefore, it is more
logical to consider Tresca criterion as a special
case of the Coulomb’s criterion rather than
Coulomb’s criterion as a generalization of Tresca
criterion.
The simplest form of dependence (1.22) of the
Mohr-Coulomb criterion is the linear form
The Mohr-Coulomb and Drucker-Prager
Criteria
The Coulomb (1776) is based on the idea of
dependence of tangential limit stresses s on
normal stresses rn
Here r1 ; r3 are the maximum and minimum
(accounting for the signs) principle stresses
r1 [ r3 . Thus, r3 , r1 are the maximum and
the minimum compressive stresses, respectively.
The representation of (1.24) differs from the one
used in the literature on rock mechanics, because
the difference in the convention of signs for
stresses leads to different numbering of the main
stresses.
Another wide-spread criterion is that of
Drucker and Prager (1952), according to which
the transition to an inelastic region occurs after a
certain combination of the critical stress tensor
invariants has been achieved.
jsj ¼ f ðrn Þ
ð1:22Þ
that essentially distinguishes it from traditional
strength criteria for metals, which depend on
shear stresses only: either the maximum shear
stress, as in the criterion of Treska (1864), or
invariant of shear stresses, as in the criterion of
Huber (1904), Von Mises (1913). It is interesting
s ¼ s0 rn tgq0
ð1:23Þ
Here s0 ; q0 are cohesion and an angle of
internal friction of the medium. Hereinafter, the
tensile stresses are considered positive, according
to the convention for stresses signs adopted in
continuum mechanics, which is opposite to
convention of the mechanics of soils and rocks.
Therefore, the forms of representation of the
criteria may be unusual: they differ from the
wide-spread form by signs before terms containing first degree of the normal stresses.
It is convenient also to write down
Mohr-Coulomb criterion in terms of principle
stresses
r3 þ
1 þ sin q0
2s0 cos q0
r1 ¼ 0 ð1:24Þ
1 sin q0
1 sin q0
f ðI 1 ; I 2 Þ ¼ 0
ð1:25Þ
8
1 Stress-Strain State of Rocks
I1 ¼ r1 þ r2 þ r3 ;
r0jk
¼
1
rjk rii djk
3
I2 ¼
1 0 0
r r ;
2 jk jk
ð1:26Þ
Here r1 ; r2 ; r3 are the principle stresses; I1 is
the first variant of the stress tensor; I2 is the
second variant of the stress deviator r0jk .
Similar to the Mohr-Coulombcriterion, a linear dependence in one of the following forms is
also widely used
ri þ aDP rm ss ¼ 0
pffiffiffiffiffiffi
1
rm ¼ ðr1 þ r2 þ r3 Þ; ri ¼ 3I2
3
ð1:27Þ
ð1:28Þ
where ri is stress intensity; rm is hydrostatic
pressure; aDP ; and ss is material constants. Or
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
pffiffiffi ðr1 r3 Þ2 þ ðr1 r2 Þ2 þ ðr2 r3 Þ2
6
þ B ð r 1 þ r2 þ r3 Þ A ¼ 0
ð1:29Þ
A; B are constants. Obviously, there is a simple linear relationship between parameters A; B
and aDP , ss . Hereafter, only the second form
(1.29) will be used.
Note that the criterion in a form similar to
(1.29) was proposed by other authors (Botkin
1940; Nadai 1950; Mirolyubov 1953), and earlier
works by Drucker-Prager, so it would be more
correct to refer to the criterion (1.29) as Botkin’s
criterion (Baklashov and Kartozia 1975).
The Mohr-Coulomb and Drucker-Prager criteria describe different surfaces in the stress space:
the first one corresponds to a hexagonal pyramid,
the second corresponds to a cone. Therefore, there
could be no unique correspondence between the
parameters of the criteria. When determining the
parameters criteria, these surfaces (pyramid and
cone) are constructed as passing through the
points of experimental data in the stress space.
Therefore, their intersection lines should pass
through the points corresponding to the experimental data. Usually two parameters for each of
the criteria are determined by more than two
experiments, and surfaces are constructed using
the least squares method, i.e. minimizing the sum
of the squares of the relative error between the
experimental and calculated (lying on the surface)
values. The surfaces constructed in this way will
intersect along some lines, but the experimental
points may no longer lie on these lines. Cases that
correspond to various typical tests or hypothetical
schemes can be used to estimate the range of
variation in the relationship between criteria
parameters. So, for Karman-type testing, r1 ¼
r2 [ r3 the relation looks like
6s0 cos q0
A ¼ pffiffiffi
;
3ð3 sin q0 Þ
2 sin q0
B ¼ pffiffiffi
3ð3 sin q0 Þ
ð1:30Þ
This combination of parameters corresponds
to the Drucker-Prager cone circumscribed the
Mohr-Coulomb pyramid.
The case r1 ¼ r2 \jr3 j corresponds to some
“average” position of the cone, and the relation
between the parameters is
6s0 cos q0
A ¼ pffiffiffi
;
3ð3 þ sin q0 Þ
2 sin q0
B ¼ pffiffiffi
3ð3 þ sin q0 Þ
ð1:31Þ
For the inscribed Drucker-Prager cone to the
Mohr-Coulomb pyramid (e.g. McLean and Addis
1990), the relationship between the parameters
has the form
3s0 cos q0
A ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
;
9 þ 3 sin2 q0
sin q0
B ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
9 þ 3 sin2 q0
ð1:32Þ
For generalized shear tests r1 [ r2 [ r3 ; r2 ¼ r1 þ2 r3 ; the relationship between the
parameters is
A ¼ s0 cos q0 ;
B¼
sin q0
3
ð1:33Þ
Usually the first (sometimes the second)
relation is used. Such a choice is conditioned by
the prevalence of experimental data obtained
with Karman-type devices. Some authors (e.g.
1.2 Transition to Inelastic Deformation
9
McLean and Addis 1990) add third (1.32) relation to provide the widest boundaries between
the parameters. The last, rarely considered
dependence (1.33) is provided not due to its
simplicity, but because of its importance for tests
carried out with TILTS (Chap. 4). We emphasize
that when calculating parameters based on a
specific set of experiments, the relationship
between the parameters of the models may differ
from those described above. It would not,
therefore, be accurate to speak of a choice from
the closed set of options presented. It appears
more correct to independently calculate the
parameters of the Drucker-Prager and
Mohr-Coulomb criteria based on the results of
experiments using the method of least squares for
each of the criterion, and then establish the
relationship between them. Such an approach is
especially attractive when using a sufficient
number of non-standard test results. However, in
any case, the discrepancies between the results
obtained with (1.30)–(1.33) are not always
essential for practical purposes, and are rather of
theoretical importance.
Note that although the parameters of the
Coulomb-Mohr criterion, in principle, are not
subject to any restrictions: theoretically, the
cohesion may vary from zero to any large value,
and the angle of internal friction lies within the
range of zero to ninety degrees, too large values
of the parameter B of Drucker-Praguere model
leads to unrealistic situations. Thus, by substipffiffiffi
tuting the value B [ 1= 3 0:577 in (1.29) we
find that such a body may not be destroyed by
uniaxial compression. The analysis shows that in
order for the body to collapse at any proportional
plane stress compression
r1 ¼ r; r2 ¼ kr;
r3 ¼ 0ðr [ 0; 0 k 1Þ
It’s necessary to meet the condition
1
B pffiffiffiffiffi 0:288
12
ð1:34Þ
This value corresponds to the most “unfavourable” value k ¼ 1 for the criterion,
corresponding to two-axis compression. However, the fulfillment of the condition (1.34) is not
strictly obligatory for practical situations,
because the criterion (1.29) is only a linearization
of the real, generally nonlinear, dependence, and
if the range of the stress under consideration does
not extend the range in which non-physical
artifacts of the model may appear, there should
be no contradiction.
The results obtained using the Coulomb-More
(1.24) and Drucker-Prager (1.29) criteria are
close, and the choice is determined mainly by
convenience rather than accuracy. The
Mohr-Coulomb criterion is more appropriate for
the analysis of simple situations. The
Drucker-Prager criterion is more appropriate for
calculations of complex non-equal stress states,
since the values of the three main stresses appear
in it interchangeably and there is no need to rank
them beforehand.
When parameters q0 and B vanish, criteria
(1.24), (1.29) are reduced to Treska and
Huber-Mises criteria, respectively.
Other Criteria
More precise, though more time-consuming at
the stages of both obtaining and applying, is the
non linearized Mohr-Coulomb criterion, according to which the dependence of the critical shear
stress on normal stress (1.22) is obtained by
constructing the envelope of the critical Mohr
circles for individual experiments. This curve is
called the strength passport, and its application
and method of obtaining were laid down in the
national standard [GOST 21153.8–88].
At present, the Hoek-Brown criterion is
widely used (Hoek and Brown 1980), which may
be written in the form of the following empirical
expression
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a
r1 ¼ r3 þ AHB ðr3 Þ þ B2HB
ð1:35Þ
Here r1 ; r3 are the maximum and minimum
main compressive stresses; a; AHB ; BHB are
model parameters, which in turn may be obtained
with the help of empirical dependences (usually
it is supposed a ¼ 2Þ rather than from the direct
10
experiments. Obviously, this criterion is a particular type of criterion (1.22).
The peculiarity of this criterion comparing to
the traditional Mohr-Coulomb criterion lies in its
nonlinearity that better describes the behavior of
real rocks, which is, of cause, a positive point.
However, the type of nonlinearity is prescribed
by formula (1.35), and it is not always clear what
advantages the prescribed dependence on a particular type of empirical nature possesses comparing to a dependence that can be obtained from
a set of direct tests. Apparently, the main purpose
of using this criterion is to promote the ideology
of reduction the number of direct tests and the
wider use of correlations, analogies, similarities,
and empirical corrections to varying conditions.
A detailed analysis of the advantages and disadvantages of this model has been repeatedly
discussed (e.g., Sas and Bershov 2015) and is
beyond the scope of the current study.
As drawbacks of the Mohr-Coulomb and
Drucker-Prager criteria, it was noted that it was
impossible to describe failure of materials in the
state of hydrostatic compression or the close
states. Indeed, according to these criteria, for
any shear stress (or for any shear stress intensity) there is a sufficiently high normal stress
(hydrostatic compression) at which failure will
not occur. Experiments show, however, that
failure under such loads can still happened. Two
types of criteria were proposed to describe these
processes. The first group consists of cap models according to those the curve of the critical
state in the plane of normal and shear stresses in
the domain of compression does not tend to
infinity, but returns to the abscissa axis corresponding to normal stresses (Sandler et al. 1976;
Schwer and Murray 1994). The second group
includes the Cam-clay model (Roscoe et al.
1968), according to those the limit curve in the
same plane is a pair of lines connected by a
circle, or a circle touching the zero-zero point
and lying in the domain of compressive stresses.
Consideration of these models is beyond the
scope of the current study.
Note that neither the Mohr-Coulomb criterion
in its classical form (1.23), nor its more complex
variant, the “strength certificate”, nor the
1 Stress-Strain State of Rocks
Hoek-Brown criterion, in volve the intermediate
principle stress (Ishlinskiy 1954). Although these
criteria are based on a huge amount of experimental data, the absence of one of the parameters
raises concerns among a number of researchers.
Although Drucker-Prager criterion contains the
values of all three main stresses, the type of
dependence on the intermediate stress is prescribed, and doubts about the universality of this
dependence are of the same nature as doubts
about the adequacy of the absence of the
dependence in Mohr-Coulomb criterion. Concern
on the subject has been mobilized a number of
researchers, probably starting with Moggy, on the
detailed study of the influence of the intermediate
principle stress on the strength of rocks. For this
purpose, the first true three-axial testing devices
were created. At present, the study of the influence
of intermediate principle stress on rock strength
has become a kind of trend (Murrell 1963; Mogi
1966, 1967, 1971; Chang and Haimson 2000;
Haimson and Chang 2002; Haimson 2006, 2007;
Haimson and Rudnicki 2009; Haimson et al.
2010). However, the results show that the value of
the intermediate principal stress, although has a
systematic effect on the strength, especially at
close to zero values of the minimum principal
stress, is not decisive for isotropic rocks.
As noted above, the presence of structure and
texture in rocks leads to the appearance of anisotropy of physical and mechanical properties,
many of which have a tensor nature and are
determined by the tensors of the corresponding
ranks. However, a number of important properties, first of all strength, do not possess tensor
nature. The non-tensor nature of the dependence
of strength on the direction of acting stresses is
directly observed in experiments, for example,
(Mogi 1971; Singh et al. 2015; Karev et al.
2016). To describe the dependence of the
strength of anisotropic rocks on applied stresses a
number of criteria were proposed (see, for
example, (Barton 1971, 1976, 2011; Goodman
1980; Hoek and Brown 1997; Singh et al. and the
reviews provided in that works).
The simplest generalization of the
Mohr-Coulomb criterion describing the strength
of anisotropic rocks was proposed by Jeager
1.2 Transition to Inelastic Deformation
11
(1960), see also Goodman (1980). The idea
consists in supposing that the cohesion and
internal friction are direction-dependent. It was
noted that the dependence for cohesion is more
significant than the dependence for the internal
friction angle (ibid.).
A Criterion Based on Two Mechanisms of
Fracture
The non-tensor nature of the dependence of
strength (or, more broadly, the transition to the
inelastic deformation) of rocks on the direction of
stress is caused by the presence of at least two
fracture mechanisms: fracture in planes where
the combination of normal and tangential stresses
exceeds its limit, and fracture in the weakened
planes associated with the layered structure. Such
model was considered, for example, in Jaeger
(2007), Karev et al. (2016).
For a system of weakened planes with normal
n, the most suitable fracture criterion is the
Coulomb type criterion (Coulomb 1776; Barton
1971; Goodman 1980)
jsj ¼ f ðrn Þ
ð1:36Þ
Here s and rn are the shear and normal
stresses acting in theplane with normal n. The
line a approximation of (1.36) is
s ¼ sc rn tgqc
ð1:37Þ
Here sc ; qc are the adhesion and friction angle
of the weakened planes.
Normal rn and shear s stresses are expressed
through the components of the stress tensor in the
laboratory coordinate system as follows (e.g.
Kachanov 1971)
rn ¼ rij ni nj
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s ¼ p21 þ p22 þ p23 r2n
ð1:38Þ
ð1:39Þ
pi ¼ rij nj
ð1:40Þ
Here pi is the components of the traction vector
acting in the (weakened) plane with normal n; ni is
the direction cosines of the normal in the laboratory coordinate system. In the coordinate system
related to the principle stresses, the criterion
(1.37) is expressed through their values by successive substitution (1.38)–(1.40) into (1.37)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
r21 n21 þ r22 n22 þ r23 n23 r1 n21 þ r2 n22 þ r3 n23
¼ sc r1 n21 þ r2 n22 þ r3 n23 tgqc
ð1:41Þ
The competitive failure mechanism is the failure
in the planes for which the combination of normal
and shear stress s þ rn tgqc exceeds the critical
value. This mechanism is realized if a combination
of stresses along the weakened planes is insufficient to initiate fracture (e.g., for compression
normal to layering). As a criterion describing this
mechanism, it is logical to use the Mohr-Coulomb
(1.24) or the Drucker-Prager (1.29) criteria.
According to the considered model, the
elastic-plastic transition occurs by the mechanisms for which the fracture conditions are
achieved earlier: either by the weakening planes
(1.38), or by the planes corresponding to the
most unfavorable combination of shear and
normal stresses (1.24) or (1.29).
The model considered is certainly idealized,
since it assumes discrete switching of fracture
mechanisms. However, it is obvious that both
mechanisms can be realized with a certain
probability near the switching condition,
depending on the random distribution of
strength, deflection of the layering angle from the
average value, etc. Moreover, interaction of
mechanisms is possible if one of them is realized
sequentially at the micro level and some hybrid is
observed at the macro level.
12
1 Stress-Strain State of Rocks
Hill’s Criteria and Its Modifications
For anisotropic (orthotropic) materials the criterion of transition to plasticity was proposed by
Hill (1948) (see also Hill 1983; Malinin 1975).
Written in the coordinate system associated with
the orthotropy axes, it has the form
2
2
2
H
H
F H ¼ GH
ð23Þ ðr22 r33 Þ þ Gð13Þ ðr11 r33 Þ þ Gð12Þ ðr11 r22 Þ
2
H
2
H
2
2
þ 2LH
ð23Þ r23 þ 2Lð13Þ r31 þ 2Lð12Þ r12 ry [ 0
ð1:42Þ
Here rY is the yield stress. Constants LH
ðijÞ are
expressed through yield strength in each
direction
GH
ð12Þ
1
1
1
þ
;
r2Y 22 r2Y 33 r2Y 11
r2Y
1
1
1
¼
þ
;
2 r2Y 33 r2Y 11 r2Y 22
!
r2
1
1
1
;
¼ Y
þ
2 r2Y 11 r2Y 22 r2y 33
GH
ð23Þ ¼
GH
ð13Þ
r2Y
LH
ð23Þ ¼
LH
ð13Þ ¼
2
r2Y
2r2Y 23
r2Y
;
2r2Y 31
LH
ð12Þ ¼
r2Y
2r2Y 12
ð1:43Þ
Values with matching indexes correspond to
the tensile yield strength along the corresponding
axes, values with different indexes correspond to
the shear yield strength. This criterion contains 6
parameters (the parameter rY is introduced for
convenience and will be excluded from the final
expression). The criterion allows to describe the
variation in transition to inelasticity at different
angles of load application, exactly—for the
directions corresponding to the axes of orthotropic (by means of parameters GH
ðijÞ ) and direc-
tions, located at 45 to them (by means of
parameters LH
ðijÞ ), and with the aid of smooth
approximating curves, prescribed by the type of
expression (1.42), for any other direction.
Criterion (1.42) is the most general form of
quadratic form for the components of the stress
deviator. The quadratic form of an even more
general form (recorded relative to the component
of the full stress tensor) was proposed earlier by
Von Mises (1913), but, as Hill (1948) has
shown, in the absence of the influence of comprehensive compression, the number of constants
in the form proposed by Mises is excessive.
For a transversally isotropic medium with the
plane of isotropy with normal n3 the number of
parameters in (1.42) is reduced to three
r2Y 1
H
GH
;
ð13Þ ¼ Gð23Þ ¼
2 r2Y 33
r2Y
2
1
H
Gð12Þ ¼
;
2 r2Y 11 r2Y 33
LH
ð13Þ
¼
GH
ð12Þ ¼
LH
ð23Þ
r2
¼ 2Y ;
2rY 13
ð1:44Þ
r2Y
H
¼ GH
ð13Þ þ 2Gð12Þ
2r2Y 12
Generalization of Hill’s Criterion in the form
of Lui-Huang-Stout (LHS)
Anatural generalization for this law is to add a
term to the expression (1.42) that takes into
account the effect of normal stresses on the value
of the critical shear stress (Lui et al. 1997). For
consistency with Drucker-Prager criterion (1.29),
it is natural to apply square root to the parts of the
expression (1.42) before adding these terms; it is
also possible to set ry ¼ 1 without violating the
generality:
h
F L ¼ GLð23Þ ðr22 r33 Þ2 þ GLð13Þ ðr11 r33 Þ2
þ GLð12Þ ðr11 r22 Þ2 þ 2LLð23Þ r223 þ 2LLð13Þ r231 þ 2LLð12Þ r212
i1=2
þ BLð1Þ r11 þ BLð2Þ r22 þ BLð3Þ r33 1 ¼ 0
ð1:45Þ
Here rij are the components of the stress
tensor in the coordinate system associated with
the axes of isotropy of the material (presumably
having at least three mutually perpendicular axes
of symmetry of the fourth order); GLðijÞ ; LLðijÞ ; BLðiÞ
are material constants. These constants maybe
expressed through the yield strengths (or stress
corresponding to elastic-plastic transition) in
compression and tension along the corresponding
axes rCi ; rTi , and at shear rSij (Lui et al. 1997).
1.2 Transition to Inelastic Deformation
GLð23Þ þ GLð13Þ ¼
C
2
r3 þ rT3
2rC3 rT3
13
ð1:46Þ
1
LLð23Þ ¼ 2
2 rS23
ð1:47Þ
rC1 rT1
2rC1 rT1
ð1:48Þ
BLð1Þ ¼
The remaining values are obtained by cyclic
permutation of indexes. It can be seen from
(1.46) to (1.48) that all nine parameters are
independent.
However, it is preferable to determine constants GLðijÞ ; LLðijÞ ; BLðiÞ from other experiments, for
example, from unequal compression experiments
on specimens cut at various angles to the axes of
material orthotropy, as will be demonstrated
below (Chap. 8.1).
For a transversally isotropic medium with the
plane of isotropy oriented perpendicular to the
x3-axis, the number of parameters in (1.45) is
reduced to five by meeting the conditions (1.49)
GLð13Þ ¼ GLð23Þ ;
LLð13Þ ¼ LLð23Þ ;
LLð12Þ ¼ GLð13Þ þ 2GLð12Þ ;
BLð1Þ ¼ BLð2Þ
ð1:49Þ
For vanishing dependence of the critical stress
on normal stresses, criterion (1.45) is reduced to
Hill’s criterion of plasticity for orthotropic
materials (Hill 1948, 1983; Malinin 1975) (1.39).
On the other hand, for an isotropic body, i.e.
when the following conditions are satisfied
BLð1Þ
¼
BLð2Þ
¼
BLð3Þ
Generalization of Hill’s Criterion in the form
of Caddel-Raghava-Atkins (CRA)
An alternative, but no less natural, generalization
for Hill’s criterion of anisotropic plasticity consists in adding a term that takes into account the
effect of normal stresses on the value of the
critical shear stress to the expression (1.42)
(Caddel et al. 1973) (without applying the square
root to value of the equivalent shear stress)
F C ¼ GCð23Þ ðr22 r33 Þ2 þ GCð13Þ ðr11 r33 Þ2
þ GCð12Þ ðr11 r22 Þ2 þ 2LCð23Þ r223
þ 2LCð13Þ r231 þ 2LCð12Þ r212
þ BCð1Þ r11 þ BCð2Þ r22 þ BCð3Þ r33 1 ¼ 0
ð1:51Þ
Here it is also possible to set ry ¼ 1 without
violating the generality; rij are the components
of the stress tensor in the coordinate system
associated with the axes of isotropy of the
material; GCðijÞ ; LCðijÞ ; BCðiÞ are the material constants. Similarly to LHS model, these constants
may be expressed through the yield strengths (or
stress corresponding to elastic-plastic transition)
in compression and tension along the corresponding axes rCi ; rTi and shear rSij (Caddel et al.
1973).
GCð23Þ þ GCð13Þ ¼
GLð13Þ ¼ GLð23Þ ¼ GLð12Þ ¼ G;
LLð13Þ ¼ LLð23Þ ¼ LLð12Þ ¼ 3G;
GLð13Þ ¼ GLð23Þ ¼ GLð12Þ ; LLð13Þ ¼ LLð23Þ ;
LLð13Þ ¼ 3GLð12Þ ; BLð1Þ ¼ BLð2Þ ¼ BLð3Þ
ð1:50Þ
¼ B=3
the criterion (1.45) is reduced to the
Drucker-Prager criterion (Drucker and Prager
1952) (1.29).
For some layered rocks, the strength limits
along and normal to the layering are either the
same or differ slightly (Karev et al. 2016), which
leads to additional equalities
1
rC3 rT3
ð1:52Þ
1
LCð23Þ ¼ 2
2 rS23
ð1:53Þ
rC1 rT1
rC1 rT1
ð1:54Þ
BCð1Þ ¼
The remaining values are obtained by cyclic
permutation of the indexes. Note, the formulas
LCðijÞ for both criteria (LHS and CRA) are the
same, the values BCðiÞ calculated using the
14
1 Stress-Strain State of Rocks
formulas (1.48) and (1.54) differ by half. All nine
parameters are still independent.
Similar to Lui-Huang-Stout model (1.45), for
a transversally isotropic medium with an oriented
isotropy plane perpendicular to x3-axis, the
number of parameters is reduced to five due to
fulfillment of conditions (1.49), and to three
(1.50) for isotropic medium.
In contrast to the criterion (1.45), which is
reduced to the Drucker-Prager criterion in the
case of isotropy, for the criterion (1.51) the
relationship between the tangent stress intensity
ri and the first invariant (hydrostatic stress) 3rm
becomes nonlinear
ri ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
arm þ b
ð1:55Þ
where a; b are constants. The non-linearity of the
formula (1.55) itself should not be considered as
a disadvantage. Nor is the lack of transition to the
linear formulation of Drucker-Prager criterion a
serious drawback. Moreover, the type of dependence (1.55) assumes a slower growth of the
ultimate intensity of the tangential stresses with
the growth of the compression than the linear
one, which seems to be more consistent with the
reality. For transition to isotropy (1.55) the criterion (1.51) becomes somewhat analogous to
Hawkeyek-Brown criterion, differing from that
criterion the same way as the Drucker-Prager
criterion differs from the Coulomb-More criterion, i.e. the criterion (1.55) relates stress
invariants rather than the principle values.
In (Deshpande et al. 2001), a similar modification of the Hill criterion was proposed to take into
account the effect of comprehensive compression
F ¼ Gð23Þ ðr22 r33 Þ2 þ Gð13Þ ðr11 r33 Þ2
þ Gð12Þ ðr11 r22 Þ2
þ 2Lð23Þ r223 þ 2Lð13Þ r231 þ 2Lð12Þ r212
þ Bðr11 þ r22 þ r33 Þ2 1 ¼ 0
ð1:56Þ
The difference from (1.51) here is that normal
stresses are included in this expression rather
with the second rather than first power. In addition, instead of three different coefficients at
normal stresses, only one coefficient at the square
or their sum is introduced. Some analogue of this
criterion can be obtained by setting
Bð1Þ ¼ Bð2Þ ¼ Bð3Þ ¼ B=3
ð1:57Þ
in (1.51).
According to (1.48), this condition cannot be
implemented for arbitrary material with different
yield stresses on mutually orthogonal axes, and
this is a significant, in some cases critical disadvantage. However, due to the phenomenological nature of the criterion, this condition,
although slightly reducing the accuracy, allows
reducing the number of constants. Moreover,
even without using the limitation imposed by the
condition (1.57), the entire variety of experimental data (e.g. Singh et al. 2015) cannot be
accurately described within the framework of the
criteria of type (1.45) with nine parameters.
Therefore, the additional reduction the number of
model parameters by two formula (1.57) does not
change the qualitative character of the approximation used.
Modifications of Hill’s criterion similar to
(1.51) have been proposed in (Shih and Lee
1978; Valliappan et al. 1976) aiming at taking
into account the difference in tensile and compressive yield strength. That criteria did not
contain a radical sign and imposed a restriction
on coefficients playing the roles of coefficients Bi
that consists in requirement of their sum vanishing; this restriction was introduced to ensure
incompressibility using the criterion as a potential for the associated law of plastic flow. When a
similar criterion is used to describe fracture (or
transition to inelasticity) of such materials as
rocks (as well as soils, concretes, ceramics,
composites), this restriction is not physically
justified.
1.2 Transition to Inelastic Deformation
15
On the Relationship Between Parameters of
Generalized Hill’s Criterion in the Forms of
Caddel-Raghava-Atkins and Lui-HuangStaut
If we assume that the intersection lines of the
surfaces in space of stresses defined by criteria
CRA and LHS correspond touniaxial compression, uniaxial tension along the directions of axes
of symmetry, and also pure shear, then comparison (1.46)–(1.48) with (1.52)–(1.54) yields
GCð12Þ
BL BL BL
¼ GLð12Þ þ 3 1 2
2
2
2
ð1:58Þ
LCð23Þ ¼ LLð23Þ
ð1:59Þ
BCð1Þ ¼ 2BLð1Þ
ð1:60Þ
with cyclic permutation of the indexes. However,
since the surfaces defined by the criteria do not
coincide, it is preferable to determine parameters
of each criterion independently from the available experiments.
Restrictions to be Imposed on the Parameters of the Generalized Hill Criterion in the
Forms of Caddel-Ragava-Atkins and Lui
Huan-Staut for Rocks
For a transversally isotropic medium, the generalized Hill criterion in the form of a
Lui-Huang-Staut (1.45) is characterized by (1.49)
five parameters
GLð13Þ ;
GLð12Þ ;
LLð13Þ ;
BLð1Þ ;
BLð3Þ
ð1:61Þ
The remaining four are expressed in terms of
them as follows
G:ð23Þ ¼ GLð13Þ ; LLð23Þ ¼ LLð13Þ ;
ð1:62Þ
LLð12Þ ¼ GLð13Þ þ 2GLð12Þ ; BLð2Þ ¼ BLð1Þ
Like the constraints (1.34) on the constant
B of Drucker-Prager criterion, it is possible to
obtain constraints on constants BLð1Þ ; BLð3Þ in the
generalized Hill’s criterion. Thus, it is followed
from (1.45) that for rocks that could reach the
critical state in uniaxial compression applied
along and normally to the plane of isotropy, the
following conditions should be satisfied
BLð1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
GLð12Þ þ GLð13Þ
ð1:63Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffi
2GLð13Þ
ð1:64Þ
BLð3Þ For realizing the possibility of reaching the
critical state at biaxial compression applied in
two directions in the plane of isotropy and in the
plane normal to the plane of isotropy, it follows
from (1.45) that
BLð1Þ sffiffiffiffiffiffiffiffiffiffi
GLð13Þ
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
BLð1Þ þ BLð3Þ GLð12Þ þ GLð13Þ
ð1:65Þ
ð1:66Þ
Obviously, restrictions (1.65), (1.66) are
stronger than criteria (1.63), (1.64).
Constraints (1.63)–(1.66) are not rigorous and
may be violated. Necessity of their fulfillment
and possibilities of their violation remain the
same as for the constraint on the parameters of
the Drucker-Prager criterion.
In addition to these restrictions for transversally isotropic rocks, there are restrictions of
another nature. For the majority of sedimentary,
metamorphic and some effusive igneous rocks
the planes of isotropy (planes of layering) coincide with the planes of weakening. That is why
the compressive strength of such rocks along to
the normal to layering, rC3 usually at least not less
than the compressive strength in the plane of
layering, rC1 . The values of the tensile strength is
reverse: the tensile strength along the normal to
layering, rT3 , is usually not more than the tensile
strength in the plane of layering, rT1 . The shear
strength within the planes of layering, rS12 , is
usually not less than the shear strength within the
planes normal to the plane of layering, rS13
16
1 Stress-Strain State of Rocks
rC3 rC1 ;
rT3 rT1 ;
rS12 rS13
ð1:67Þ
When substituting these inequalities into
(1.48) the following inequality is obtained
BLð3Þ
BLð1Þ
ð1:68Þ
At the same time, it is not possible to obtain
similar inequality for G0ð13Þ ; G0ð12Þ , because the
inequalities in the first two formulas (1.67) are
differently directed. Numerical analysis shows
that the relation of constants BLð3Þ ; BLð1Þ gives a
more noticeable asymmetry in the fracture in
compression along different directions than the
relation of constants GLð13Þ ; GLð12Þ .
From the last relation (1.67), the third formula
(1.62) and the relation (1.47), it follows that for
rocks the following inequality
LLð13Þ GLð13Þ þ 2GLð12Þ
ð1:69Þ
should be satisfied, the ratio
LLð13Þ
GLð13Þ þ 2GLð12Þ
1
ð1:70Þ
characterizing the magnitude of the “drop” of the
critical compressive stress comparing to the
critical compressive stress in the direction of
layering.
Since according to the Caddel-Ragava-Atkins
criterion, with the growth of the absolute value of
the applied stresses, the contribution of shear
stresses grows faster than the contribution of
normal stresses, no restrictions of the kind (1.34)
or (1.63)–(1.66) should be imposed on the model
parameters.
However, for restrictions related to the nature
of rocks—the presence of a system of weakened
planes—the restrictions remain: thus, substituting
(1.52), (1.54) into the first two formulas (1.67),
we obtain
BCð3Þ
2GCð13Þ
BCð1Þ
GCð13Þ þ GCð12Þ
ð1:71Þ
It is naturally to suppose, that the difference in
the critical tensile stresses along and normal to
layering (second formula 1.67) is more pronounced than the difference in the critical compressive stresses (first formula). Thus, setting
1
1
1
1
C C
rT3 rT1
r1 r3
ð1:72Þ
is followed by inequality
BCð3Þ BCð1Þ
ð1:73Þ
similar to the one determined (1.68).
The ratio similar to the one defined by (1.70)
with the replacement of the index L by C have
the same meaning: it determines the magnitude
of the “drop” of the critical compressive stress
comparing to the critical compressive stress in
the direction of layering.
It should be emphasized once again that the
inequalities obtained in this paragraph, similar to
those for the Drucker-Prager criterion, are not
rigorous, since it is desirable to determine the
parameters in such a way that the criteria
describe the behavior of rocks in the range of
interest, extrapolation to the range of tensile
stress (usually not too interesting from the point
of geomechanic applications) may lead to deviations from the realistic description.
Deformation
on
Reaching
Critical
Condition
Once the critical state of the rock has been
reached, rocks usually do not fail instantly, but
continues to deform. The deformation at that
stage has a sufficiently large inelastic component.
Such behavior leads researchers to a natural
desire to use theory of plasticity, and as its
mostly complete and developed variant—the
1.2 Transition to Inelastic Deformation
theory of plastic flow, to describe inelastic
deformation of rocks (Hill 1983; Malinin 1975).
The development of the theory of plastic flow to
take into account specific phenomena inherent in
rocks has been done by many authors (Drucker
and Prager 1952; Nikolaevsky 1967, 1996;
Lomakin 1980, 1991; Chanyshev 1984; Morita
and Grary; Stefanov 2005; Karev et al. 2016;
Ustinov 2016). Consider a variant of such a
theory, accounting the key features of rock
deformation mentioned above Sect. 1.2).
On exceeding the stresses corresponding to
the initial yield surface, or more precisely—criterion of elastic-inelastic transition, in our case
criterion (1.45), plastic strains appear in addition
to elastic strains. When unloading the strains
have only an elastic component. When reloaded,
plastic strains appear only when the stresses
reach the maximum level achieved in previous
cycles. Therefore, we can say that the criterion of
elastic-plastic transition is a function of the
maximum achieved stresses. The problem of
evolution of the critical stresses during repeated
loading along the path different from the path of
previous loading requires separate consideration.
In this case, the criteria can be described as a
yielding surface changing under the influence of
achieved stresses. The character of this change
may vary. Thus, under the action of the stresses
of the opposite sing as compared to the previous
loading, both the increase and decrease of the
absolute value of the critical stress may be
observed [Baushinger effect (Nadai)]. Based on
the experience of generalizing a huge amount of
experimental data, it can be concluded that the
law of transforming of the yield surface is a
property of the material, and a particular
approximation, suitable for describing the material or class of materials under consideration,
should be chosen according to the observed
mechanical behavior. The extreme variants of the
law of the yield surface transformation are:
(i) isotropic hardening, according to which the
yield surface expands in a similar way in the
stress space in all directions; and (ii) translation
hardening, according to which the yield surface
shifts in the stress space, preserving its shape and
17
size (Hill 1983; Malinin 1975). In any case, it is
assumed that the condition of belonging the
current combination of stresses to the yield surface is met for active loading.
For an isotropic body, assuming the dependence of the yield surface only on the achieved
stress state, the yield surface may be written as a
function of the principle stresses r1 ; r2 ; r3
F ðr1 ; r2 ; r3 ; ki Þ ¼ 0
ð1:74Þ
or their combinations, such as invariants. In
general, the yield surface F may also contain a
number of parameters ki .
For the anisotropic media under consideration, it is natural to accept some generalization of
criterion of elastic-inelastic transition as a yield
surface. Among criteria of Mohr-Coulomb and
Drucker-Prager types, preference should be
given to the second (which is usually done),
because theories of inelastic deformation based
on the criteria of Mohr-Coulomb type is associated with two groups of difficulties. The first
group is related to the need to rank the values of
the main stresses and to distinguish the maximum and minimum principle stresses; the second
group of difficulties is related to the consideration
of deformation for stress combination corresponding to angular points of the yield surface.
Let us write the expression for the yield surface of a sufficiently general type in the form of a
generalization of criterion (1.45)
n
F L ¼ GLð23Þ ½ðr22 a22 Þ ðr33 a33 Þ2
þ GLð13Þ ½ðr11 a11 Þ ðr33 a33 Þ2
þ GLð12Þ ½ðr11 a11 Þ ðr22 a22 Þ2
þ 2LLð23Þ ðr23 a23 Þ2 þ 2LLð13Þ ðr13 a13 Þ2
o1=2
2
þ 2L:L
ð
r
a
Þ
12
12
ð12Þ
þ BLð1Þ ðr11 a11 ÞBLð2Þ ðr22 a22 Þ
þ BLð3Þ ðr22 a22 Þ AðkÞ ¼ 0
ð1:75Þ
or a generalization of criterion (1.51)
18
1 Stress-Strain State of Rocks
F C ¼ GCð23Þ ½ðr22 a22 Þ ðr33 a33 Þ2
þ GCð13Þ ½ðr11 a11 Þ ðr33 a33 Þ2
þ GCð12Þ ½ðr11 a11 Þ ðr22 a22 Þ2
þ 2LCð23Þ ðr23 a23 Þ2
þ 2LCð13Þ ðr13 a13 Þ2 þ 2LCð12Þ ðr12 a12 Þ2
þ BCð1Þ ðr11 a11 Þ þ BCð2Þ ðr22 a22 Þ
þ BCð3Þ ðr22 a22 Þ AðkÞ ¼ 0
ð1:76Þ
The letter differs from the former by the
absence of the radical sign, A certain yield
strength A is introduced here. The yield surfaces
(1.75), (1.76) contain combinations of translational and isotropic hardening in the spirit of
Kadashevich and Novozhilov (1958), (see also
Malinin 1975). Isotropic hardening is controlled
by the change of the parameter A, translational
hardening is controlled by the displacement of
the center of the yield surface in the stress space,
determined by the coordinates aij , also may be
called shift stresses. Purely translational hardening and purely isotropic hardening are obtained
as individual cases.
Under active loading the growth of stresses is
accompanied by the growth of plastic deformations, which is described by the law of plastic
flow. An essential feature of the applicable law,
as noted above, is its non-associativity, i.e. its
representation in the form of
dePij ¼ dk
@Q
@rij
ð1:77Þ
Here is dk an unknown coefficient [not to be
confused with the Lamé constant k (1.6)]; depij are
increments (“rates”) of plastic deformations; Q is
a plastic potential, i.e. some function that does
not coincide with the function of yield surface F,
for the definition of which it is necessary to
introduce additional assumptions. Within the
framework of the classical variant of the plastic
flow theory, the equating of the plastic potential
with the yield function Q ¼ F allows to obtain
and justify an elegant derivation of the defining
relation of constitutive law of the theory of
plastic flow, which quantitatively describe the
inelastic behavior of metals (primarily iron
alloys) for a wide range of complex loading
programs (Hill 1983).
This choice of plastic potential equal to yield
function is called the associate flow rule because
it associates the potential with the yield function.
Starting with Drucker and Prager (1952), associate law was used to describe inelastic deformation of rocks and soils at aij 6¼ 0 (Morita,
Grary; Lui et al. 1997). However, the use of the
associated flow rule for rocks and other media
with non-vanishing volumetric inelastic strains,
leads to strongly overestimated values of inelastic volumetric deformations compared to the
observed values. In order to eliminate this discrepancy it was proposed to use non-associate
flow rule Q 6¼ F (Nikolaevsky 1967, 1996).
To adequately describe the volume inelastic
strains, the concept of dilatancy was proposed by
Reynolds (1885), according to which the volume
inelastic strains are not determined by volume
stresses, but depend on the intensity of inelastic
shear strains. Accepting the spirit of this concept
it is natural to suppose that for the deviator part
of the inelastic strain all statements of the classical theory of plastic flow, including the associate flow rule, remain valid, and for the
volumetric part of the inelastic strain, an additional law is introduced that relates increments of
volumetric and shear strains.
Based on the above, we will accept the form
of a plastic potential completely similar to the
form of the yield surface (1.75), or (1.76), with
the difference only in coefficients of linear stress
terms (which will ensure the preservation of
associativity for the deviator part of the inelastic
deformation increment).
1.2 Transition to Inelastic Deformation
19
n
QL ¼ GLð23Þ ½ðr22 a22 Þ ðr33 a33 Þ2
þ GLð13Þ ½ðr11 a11 Þ ðr33 a33 Þ2
þ GLð12Þ ½ðr11 a11 Þ ðr22 a22 Þ2
þ 2LLð23Þ ðr23 a23 Þ2 þ 2LLð13Þ ðr13 a13 Þ2
o1=2
2
þ 2L:L
ð12Þ ðr12 a12 Þ
LQ
þ BLQ
ð1Þ ðr11 a11 Þ þ Bð2Þ ðr22 a22 Þ
þ BLQ
ð3Þ ðr22 a22 Þ
ð1:78Þ
QC ¼ GCð23Þ ½ðr22 a22 Þ ðr33 a33 Þ2
aij ¼ ga epij
þ GCð13Þ ½ðr11 a11 Þ ðr33 a33 Þ2
þ GCð12Þ ½ðr11
a11 Þ ðr22 a22 Þ
a22 Þ
ð1:79Þ
Term AðkÞ is also omitted here, since vanishes
during differentiation. In the absence of inelastic
volumetric deformations, there is a restriction
LQ
LQ
BLQ
ð1Þ þ Bð2Þ þ Bð2Þ ¼ 0
ð1:80Þ
Stronger requirement
LQ
LQ
BLQ
ð1Þ ¼ Bð2Þ ¼ Bð2Þ ¼ 0
ð1:81Þ
corresponds to the imposition of additional
requirements on the ratio of inelastic deformations under tension and compression, for example, if condition (1.57) is fulfilled.
In order to ensure “deviator associativity” it is
necessary to set
;
BLQ
ðiÞ ¼ BðiÞ B0 ;
B0 ¼
3
1X
BL
3 j¼1 ðjÞ
ð1:83Þ
Values aij ; epij are supposed to be purely
deviatory. Coefficient ga is assumed to depend on
2
þ 2LCð23Þ ðr23 a23 Þ2 þ 2LCð13Þ ðr13 a13 Þ2
þ 2LCð12Þ ðr12 a12 Þ2
CQ
þ BCQ
ð1Þ ðr11 a11 Þ þ Bð2Þ ðr22
þ BCQ
ð3Þ ðr22 a22 Þ
Similar expressions for the constants of plastic
potential (1.79) are obtained by replacing the
index L with index C in expressions (1.80)–
(1.82).
To close the system of equations, the laws of
change of hardening parameters (for the case in
question, aij and AðkÞÞ, as well as a dependence
of dk are required.
To calculate the coordinates of the yield surface center, aij , it is usually assumed that aij are
proportional to the accumulated plastic
deformations
ð1:82Þ
the intensity of the additional stresses ga ¼
qffiffiffiffiffiffiffiffiffiffiffiffi
3
(Kadashevich and Novozhilov
ga
2 aij aij
1958), or on the intensity of the active stresses
(Harutyunyan 1964; Harutyunyan and Vakulenko 1965), or, in the simplest variant, to be a
material constant (Ishlinskiy 1954).
As a parameter of isotropic hardening, either
the value of the plastic deformation work,
defined as
dk ¼ rij depij
ð1:84Þ
or the value of accumulated plastic deformations
is Odquist parameter, (Odquist 1933—Odquist F
K J Zeits. and Math. Mech. 13 360)
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 p p
dk ¼
de de
3 ij ij
ð1:85Þ
For anisotropic media, contrary to the isotropic case, the use of these parameters leads to
different results (Malinin 1975).
Let us define the expression for dk. Using the
condition of belonging of the current combination
of stresses in the stress space to the yield surface
during active loading, the condition for the function F increment may be written as follows
20
1 Stress-Strain State of Rocks
@F
@F
@F
dk ¼ 0
drij þ
daij þ
@rij
@aij
@k
Substituting the values dk and daij (1.84),
(1.83) into (1.86) leads to
@F
@F
1
@F
p
p
rij depij
drij þ
ga deij dij dekk þ
@rij
@aij
3
@k
¼0
ð1:87Þ
From here, expressing the increase of plastic
deformation through the plastic potential (1.78),
we obtain
@F
@F
@Q
1
@Q
drij þ
ga
dk dij
dk
@rij
@aij
@rij
3 @rkk
@F
@Q
rij
þ
dk ¼ 0
@k @rij
ð1:88Þ
Resolving this equation with respect to dk, we
obtain
dk ¼ @F
@rij
@F
@aij
@Q
@rij
drij
@Q
@Q
13 dij @r
rij
ga H @r
kk
ij
ð1:89Þ
Here
H¼
@F
@k
rij;i ¼ 0
ð1:86Þ
ð1:90Þ
is a characteristic of the material under consideration defined from the experiment. Functions
F; Q should be understood as F L ; QL , or F C ; QC
. Thus, for a given equations F; Q; ga ; H we
have a system of differential Eqs. (1.77), (1.83),
(1.84), (1.89) for a sufficiently general case of an
anisotropic yield surface, in general case of
non-associated law plastic flow with an anisotropic plastic potential, and combination of
translational and isotropic hardening. Together
with equilibrium Eq. (1.4)
ð1:91Þ
and boundary conditions, they make up a closed
system. It should be noted that the considered
description is rather general, not related to a
particular type of yield function F and plastic
potential Q, and differs from the classical law of
the Prandtle-Reiss flow theory (Hill 1983; Malinin 1975) only by the type of functions F and
Q. Obviously, the description for isotropic rocks
may be obtained from here as a particular case.
It should be noted that, on the one hand, being
incorporated into the majority of modern calculation systems, such equations (although for
isotropic environments) have already become a
commonplace, on the other hand, the use of
non-associate laws is still a cause for discussion,
as it can lead to difficulties in numerical implementation and some paradoxical situations.
If we assume that the yield function F changes
in accordance with the plastic potential Q and
does not depend (or weakly depends) on the
volumetric stresses, the flow rule will become
associative, but the price for this will be the
introduction of two yield functions: one for the
first transition to a plastic state, and the other to
describe the developing plasticity. The introduction of additional parameters into the model
is not required, because the condition of finding a
point in the stress space during the initial transition to plasticity on both surfaces leads to the
relation of constants
ð2Þ
ð2Þ
GðijÞ ¼ G0ðijÞ =ð1 þ B0 r0 Þ2 ; LðijÞ ¼ L0ðijÞ =ð1 þ B0 r0 Þ2
ð2Þ
BðiÞ ¼ B0ðiÞ B0 =ð1 þ B0 r0 Þ
ð1:92Þ
The constants with the upper index 2 here
refer to the modified functions F and Q. The rest
of the formulas remain the same. In other words,
such modified relations (Ustinov 2016) result in
consideration of the influence of hydrostatic
stress on the yield strength as a parameter.
1.2 Transition to Inelastic Deformation
Consideration of the first invariant as a parameter in the expression for the potential was proposed in (Lomakin 1980, 2000). However a
question remains open: which value of hydrostatic pressure need to be substituted into the
expressions for F and Q: the current one, or the
one corresponding to the moment of transition to
in elasticity. The latter way seems to be more
preferable for numerical implementation for the
stress state history close to the proportional one
due to its relative simplicity, but it unlikely to be
adequate for complex loading programs containing unloading parts.
Comments and Discussion
The above system of differential equations can be
considered as a generalization of the theory of
plastic flow for anisotropic materials, criterion of
elastic-plastic transition for which includes a
dependence on normal stresses. However, a
number of questions arise.
The first issue is related to the application of
non-associate flow rule. In constructing the traditional flow theory, the presence of a plastic
potential associated with the yield function followed from the Drucker postulate, which is the
principle of maximum work of plastic deformation (Drucker 1959). The use of a non-associate
law makes the whole theory largely phenomenological. However, in the case of anisotropy, the very criterion of inelasticity is
phenomenological (1.45). The phenomenological
nature of the criteria of that kind was been
pointed out in (Lui et al. 1997).
At the same time, the yield surfaces (1.75) or
(1.76) and plastic potentials (1.78) or (1.79) with
CQ
the appropriate choice of constants BLQ
ðiÞ or BðiÞ
are co-aligned with respect to the components of
the deviator, so it is possible to speak about
partial associativity or deviator associativity of
the flow rule under consideration. In the absence
of the volume component of inelastic strain in the
flow rule (1.75) or (1.76) and plastic potential
(1.78) or (1.79), the hydrostatic component of the
stress can be considered as a parameter (similar
21
to temperature). However, it would be more
correct to explain the misalignment of the normal
to the yield surface and plastic potential by different shear and volumetric inelastic deform
abilities, and in mathematical terms—by the
different shear and volumetric plastic modules.
A model with different (shear) plastic modules on
different axes was considered in (Valliappan
et al. 1976).
The presence in the model of a combination of
isotropic and kinematic hardening leads to the
need of defining two functions ga ; AðkÞ. In some
cases, this complication may be excessive. In this
case, the model may be reduced easily to purely
isotropic or purely kinematic hardening models.
It should also be noted that the theory of
plastic flow was developed in parallel with the
accumulation of a large amount of experimental
data on complex loading. The main purpose of
the theory was to provide a quantitative
description of the observed stress-strain relationships and prediction of mechanical behavior
for other types of loading. However, the situation
is somewhat different for rocks, because: (i) there
is no such database for the study of deformation
characteristics, especially for complex loading
programs; (ii) due to the presence of anisotropy
and the pronounced influence of normal stresses,
the number of model parameters increases significantly; (iii) there are a number of problems
(related to mining openings, excavations, wells,
etc.), extremely important from a practical point
of view, in which loading does not contain
complex trajectories.
In connection with the above, a desire arises for
engineering calculations of mechanics problems
to use a simpler model—the Hill’s anisotropic
plasticity model—in which the yield strength
would depend on the volume stress as a parameter. The equation of the yield surface coinciding
with the plastic potential for this case is obtained
from (1.75) or (1.76) by the elimination the terms
with BLQ
(or BCQ
ðiÞ ) and replacing AðkÞ with
0 ðiÞ
1 0
A k; rii ; where 3 rii is the hydrostatic stress at the
moment of transition to an inelastic state.
22
This simplification, in general, hardly has a
physical justification, but it leads to a significant
simplification of engineering problems, to the
possibility of using the arsenal of computational
methods of theory of plastic flow, including the
use of ready-made finite element programs. It is
obvious that the use of such simplified models is
justified for rather simple, close to proportional
trajectories of loading, and does not cancel the
use of more accurate models where it is
necessary.
Note, that if we use a modified Hill’s criterion
in the form (1.76) as the yield surface, which is a
polynomial of the second order of the stress
components containing not only quadratic but
also linear terms (due to BCQ
ðiÞ and aij , that determine asymmetry of the yield surface in the stress
space), after the opening of brackets in (1.76), we
can see that both groups of terms are included in
the equation equally. This suggests the use of
initial stress shifts a0ij instead of constant BCQ
ðiÞ .
A similar description was performed in (Shih and
Lee 1978).
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Methods Geomech 18:657–688
Sekerzh-Zenkovich YaI (1931) To calculation on stability
of a sheet of plywood, as an anisotropic plate (in
Russian). Proceedings of TsAGI. 76:3–26
Shih CF, Lee D (1978) Further developments in anisotropic plasticity. J Eng Mater Technol 100(3):294–302
Singh M, Samadhiya NK, Kumar A, Kumar V, Singh B
(2015) A nonlinear criterion for triaxial strength of
inherently anisotropic rocks. Rock Mech Rock Eng 48
(4):1387–1405
1 Stress-Strain State of Rocks
Treska Н (1864) Comptes Rendus Acad Sci, Paris 59:754
Tymoshenko SP, Goodyear J (1979) Theory of elasticity.
M.: Science. 560p
Ustinov KB (2016) About the application of the plastic
flow models for the description of inelastic deformation of the anisotropic rocks (in Russian). Process
Geospheres 3(7):278–287
Valliappan S, Boonlaulohr P, Lee IK (1976) Non-linear
analysis for anisotropic materials. Int J Numer Methods Eng 10(3):597–606
Von Mises R (1913) Güttingtr Nachrichien, math.-phys.
Klaase. 582p
2
Deformation and Fracture of Rocks
in the Presence of Filtration
A complete system of equations in the presence
of filtration of fluid includes equations that
describe filtration, as well as equations that
ensure the connection of filtration and deformation processes. In addition, the mechanical
equations require changes accounting the features associated with the effect of pore pressure.
These changes are taken into account in the
theory of poroelasticity developed in the works
of Terzagi (1925) and Biot (1935, 1941). The
need for changes is due to the fact that when
there is fluid in the pores, it takes on itself some
of the total stress. The rest of the stresses are
perceived by the soil skeleton, this part of the
stresses is usually called the effective stresses.
Classical equations of stress equilibrium are still
recorded for total stresses, and when recorded in
terms of effective stresses they have fictitious
forces corresponding to the influence of changes
in pore pressure. The description of the kinematic
part also undergoes a change, as an additional
parameter appears—the change in the volume of
pore space. Thus, two additional parameters
appear in the constitutive equations: the change
in the volume of pore space (kinematics), and
pore pressure (statics). The changes also affect
the description of inelastic deformation: the criteria for inelastic transition and the description of
deformation upon achieving this transition will
be written in terms of effective stresses. The
following paragraphs address these issues in
more detail.
© Springer Nature Switzerland AG 2020
V. Karev et al., Geomechanics of Oil and Gas Wells,
Advances in Oil and Gas Exploration & Production,
https://doi.org/10.1007/978-3-030-26608-0_2
2.1
Filtration in Reservoir
Permeability of rock means the ability to pass
through liquids and gases at pressure changes.
There are no absolutely impermeable rocks in
nature. However, with real, relatively small
pressure drops in oil reservoirs, many rocks as a
result of insignificant pore sizes turn out to be
practically impermeable to liquids and gases
(clay, shale, etc.).
Mathematical description offiltration in rocks is
based on Darcy’s law (Darcy 1856), which established the dependence of the liquid filtration rate on
the pressure gradient. It may be written as follows
qf ¼ k Dp
S;
g L
ð2:1Þ
where q f is volumetric flowrate, m3/s; k is permeability, m2; η is dynamic viscosity of fluid, Pas;
Dp ¼ p1 p2 is pressure drop, Pa; L is length of
the specimen of porous medium, m; S is filtration
area, m2. Permeability is defined from (2.1) as:
k¼g
qf L
;
Dp S
ð2:2Þ
The unit of permeability dimension called
Darcy (D) corresponds to the permeability of a
media, through the cross-section of 1 cm2 of
which passes 1 cm3 of the liquid with the viscosity 1 cP at a pressure drop of 1 at on the base
25
26
2 Deformation and Fracture of Rocks in the Presence of Filtration
Table 2.1 Dimension of
parameters of the Darcy
equation in different
systems of units
Equation parameters
Dimension
Permeability, k
a
cm2
D
cm3/s
cm3/s
Filtration area, S
m2
cm2
m2
Rock specimen length, L
m
cm
cm
Pressure drop, Δp
Pa
dn/cm
Dynamic viscosity of fluid, η
Pa s
dn s/cm2
2
at
cP
Oil hydromechanics
ð2:3Þ
kij
p;j
g
ð2:4Þ
The tensor of permeability is expressed
through two independent constants for a transverse isotropic medium; it can be written in the
principal axes in the following form
0
k1
0
m2
m /s
Here q f is a fluid flow volumetric density
vector, m/s; p is pore pressure, Pa. In the case of
anisotropy, the permeability is characterized by a
second rank tensor kij , the Darcy’s law is being
written as
k1
kij ¼ @ 0
0
OHa
Fluid flow rate, q f
k
q f ¼ grad p
g
0
GHS
3
of 1 cm per 1 s, in laminar mode of filtration.
The physical meaning of the dimension of permeability is the cross-sectional area of channels
of porous medium through which the filtration
flow passes.
Dimension of parameters of the Darcy equation in different systems of units is given in
Table 2.1.
The validity of the law for a wide range of
parameters is confirmed by a lot of experimental
data. Deviations from Darcy law are observed at
high flow rates when it becomes turbulent.
For an isotropic case the differential form of
Darcy’s law is
qif ¼ SI
1
0
0 A ¼ k1 dij þ ðk3 k1 Þdi3 dj3
k3
ð2:5Þ
The equation of Darcy’s law must be complemented by the continuity equation for the flow
@
div qq f þ ðmqÞ ¼ 0
dt
ð2:6Þ
Here m is the porosity of the rock; q is the
fluid density. Note that porosity is understood
here as is the effective porosity that contributes to
the filtration flow.
The derivative over time vanishes for steady
process, and Eqs. (2.6), (2.4) are reduced to
kij
p;j
g
;i
¼0
ð2:7Þ
When the permeability is independent of
pressure and isotropic, the Eq. (2.7) is simplified
to the following
Dp ¼ 0
2.2
ð2:8Þ
Equations of Poroelasticisity
The Equations of Poroelasticisity Proposed
by Khristianovich and Zheltov
This theory was developed by Khristianovich
and Zheltov (1955) for high-permeable rocks
considered as granular media in relation to the
problem of hydro fracturing. The theory is based
on the statement that stresses transmitted through
any section of porous solid are decomposed into
parts: (i) stresses transmitted through the solid
2.2 Equations of Poroelasticisity
27
phase (skeleton), called effective stresses sij ; and
(ii) a part transmitted through the fluid (liquid or
gas) pressure; only the stresses transmitted
through the rock skeleton, cause its deformation.
To determine the part of the total stresses
transmitted through the skeleton the problem of
grain interactions was considered, and (omitting
the details) the connection between total rij and
effective stresses sij and pore pressure p was
obtained in the following form
is equal to tensor of the total strain defined as a
symmetric part of the displacement gradient
rij ¼ sij ð1 dÞpdij
Taking into account the relationship between
total and effective stresses (2.9), these ratios are
transformed as follows
ð2:9Þ
Here and below; rij ; sij 0, p 0; d is a share
of the total grain surface occupied by the contacts
with other grains of rock skeleton.
If the areas of contacts between the grains are
small comparing to total grain surface ðd\\1Þ,
the stresses compressing the rock skeleton is
equal to the total rock pressure minus pore
pressure. For rocks with weak plastic grains, the
contact area between the grains can be large
ðd 1Þ and the rock pressure will be transmitted
directly through the rock skeleton.
The distribution of pressure p is considered to
be either prescribed or determined from the
solution of a stationary or non-stationary filtration problem with corresponding boundary conditions. It has to be emphasized that it is the total
stresses that are subject to the equations of
equilibrium (1.3)
rij;i þ fj ¼ 0
ð2:10Þ
For effective stresses, substitution (2.9) in
(2.10) gives
sij;i ð1 dÞp;j þ fj ¼ 0
ð2:11Þ
Hooke’s law (Lekhnitsky 1977) is written for
effective stresses
sij ¼ 2leEij þ keEkk dij
ð2:12Þ
In this case, there are no inelastic deformations, and, consequently, tensor of elastic strains
eEij ¼ eTij ¼
1
ui;j þ uj;i
2
ð2:13Þ
Substitution (2.13) into Hooke’s law (2.12)
gives
sij ¼ l ui;j þ uj;i þ kuk;k dij
ð2:14Þ
rij ¼ l ui;j þ uj;i þ kuk;k dij ð1 dÞpdij
ð2:15Þ
Finally, substitution of expressions for total
stresses (2.15) into equations of equilibrium
(2.10) or expressions for effective stresses (2.14)
into equation of equilibrium for effective stresses
(2.11) gives an analogue of Lamé equations
ðl þ kÞui;ji þ luj;ii ð1 dÞp;j þ fj ¼ 0 ð2:16Þ
Biot’s Equations of Eoroelasticity
The basic equations of poroelasticity can be
obtained in another way. As the basic kinematic
variables we will chose the complete deformations eTij defined through the displacement vector
by the formula (1.1) and the relative change in
the volume of pore space, which will be referred
to as V. The force variables (generalized forces)
corresponding to these kinematic variables
(generalized displacements) will be stresses rij
and pore pressure p.
Further, either by postulating the expression
for energy in the form of an arbitrary quadratic
form of the introduced kinematic variables followed by variation, or by direct postulating the
linear relationship between the kinematic and
corresponding to them static variables, we obtain
the constitutive equations
28
2 Deformation and Fracture of Rocks in the Presence of Filtration
rij ¼ l0 ui;j þ uj;i þ k0 uk;k dij k2 Vdij
p ¼ k2 uk;k þ k1 V
Here the equality of coefficients (at Vdij in the
first equation and at uk;k in the second equation)
follows the assumption of existence of energy
potential; the minus sign is chosen in accordance
to convention of signs (positive pressure corresponds to negative stresses); constants l0 ; k0 differ, generally, from constants l; k in the previous
equations.
The total stresses rij in constitutive Eq. (2.17)
must satisfy equilibrium equation; pressure p remains independent and can be determined from
Eq. (2.8) with appropriate boundary conditions.
Substitution the second equation of (2.17) into
the first one allows excluding the change in the
pore volume V from the constitutive equations
rij ¼ l ui;j þ uj;i þ kuk;k dij aP pdij
ð2:18Þ
where
l ¼ l0 ;
k ¼ k0 k22
;
k1
aP ¼
k2
k1
ð2:19Þ
Introduction of effective stresses sij as
rij ¼ sij aP pdij
ð2:20Þ
allows to obtain for them from (2.18) constitutive
equations in the form of (2.12). Therefore formally introduced effective stresses sij have the
meaning of the part of the total stresses transmitted by rock skeleton.
Finally, substitution (2.18) into the equation
of equilibrium (2.10) leads to equations, which is
analogous to Lamé’s equations
ðl þ kÞui;ji þ luj;ii aP p;j þ fj ¼ 0
aP ¼ 1 d
ð2:17Þ
ð2:21Þ
Comparison (2.21) and (2.11), as well as
(2.20) and (2.9), gives the relation between the
parameters aP and d
ð2:22Þ
Thus, we see that both considerations of the
poroelasticity problem lead to formally the same
result.
In the latter formulation, the system is supplemented by Eq. (2.17) that allow determining
volume change V, which is not always important
for practice, but may have theoretical value.
Indeed, if we suppose that permeability may
depend on the change in pores volume
k ¼ kðVÞ
ð2:23Þ
then equation of filtration (2.7) for isotropic case
should be written as
kðVÞ
p;i
g
;i
¼0
ð2:24Þ
If dependence (2.23) is essential, the problem
becomes coupled and nonlinear (due to this
dependence), and cannot be solved sequentially
for filtration and elasticity. The complete system
of equations for this case thus includes three
scalar Eq. (2.21), the second equation of (2.17)
and Eq. (2.24) with respect to five unknowns—
three components of the displacement vector ui ,
changes in pore volume V and pressure p.
If dependence (2.23) is not essential,
kðV Þ ¼ k0 , the system becomes uncoupled: the
value of V, if of interest, may be found after
solving the problem.
In the presence of anisotropy, the system of
the equations of elasticity becomes somewhat
more complicated, not only because of the
appearance of the (fourth rank) tensor of elasticity in an explicit form in the constitutive
equations, but also because the Biot’s constant aP
should be considered as a tensor value aPij .
Indeed, in Eq. (2.19) the stress (second rank
tensor) depends on deformation (second rank
tensor) and pressure (scalar). The general
dependence of a second rank tensor (stress in our
case) and a scalar (pressure in our case) is a
second rank tensor. Therefore, in general case
2.2 Equations of Poroelasticisity
29
Biot’s constant aPij should be a second rank
tensor.
For isotropy, the second rank tensor describing an arbitrary property is represented as a
product of a constant and the unit tensor (as it is
the case in Eq. 2.19). Accordingly, for the case of
anisotropy of the Eqs. (2.19), (2.21) transform to
eTij as the sum of governed by Hooke’s law elastic
strains, eEij , and inelastic strains ePij (De Witt 1970,
1973) in the form of (1.1)
rij ¼ Kijkl uk;l aPij p
(in general, we can talk about inelastic distortion, the symmetric part of which is an
inelastic strain and the asymmetric part is an
inelastic rotation, but for the problems under
consideration for inelastic distortion we can only
consider the symmetric part (DeWit 1970, 1973).
In the case of isotropy, both elastic properties
and thermal expansion, inelastic deformations are
related to temperature DT changes as
Kijkl uk;lj aPij p;j þ fj ¼ 0
ð2:25Þ
ð2:26Þ
If the medium in question possesses
transversal isotropy, the tensor Biot’s constant is
expressed through two scalar constants aP1 ; aP3 , in
the coordinate frame associated with the axes of
isotropy of the medium
0
aP1
P
@
aij ¼
0
0
0
aP1
0
1
0
0 A ¼ aP1 dij þ aP3 aP1 di3 dj3
aP3
ð2:27Þ
Analogy of Systems of the Equations of
Poro-Elasticity and Thermo-Elasticity
The theories of thermo-elasticity and poroelasticity developed independently. The former
originated from the works of Duhamel (1837,
1838) and Neumann (1885), the latter originated
from the works of Terzagi (1925), and developed
in the works of Biot (1935, 1941). In comparison
with the classical theory of elasticity in systems
of the equations of thermo- and poro-elasticity at
least on one additional variable appears: temperature or pressure, respectively. Although the
analogy between the two theories has been
repeatedly emphasized, this analogy is not so
complete that the closed systems of equations
and boundary conditions be reduced to one
another by a simple redefinition of symbols. To
identify this analogy consider the basic equations
of thermo-elasticity.
Equations of Thermo-Elasticity
The system of equations of uncoupled
thermo-elasticity for small strains may be written
in a form reflecting the idea of decomposition,
i.e. the possibility of presenting complete strains
eTij ¼ eEij þ ePij
ePij ¼
aT
DTdij
3
ð2:28Þ
ð2:29Þ
where aT =3 is coefficient of linear thermal
expansion; aT is coefficient of volumetric thermal
expansion; eEij is elastic deformations are related
to stresses, rij , generalized by Hooke’s law (1.6)
rij ¼ 2leEij þ keEkk dij
ð2:30Þ
The components of stress or, rij , are interrelated by equilibrium Eq. (1.3)
rij;i þ fj ¼ 0
ð2:31Þ
Fora given distribution of the temperature
field, Eqs. (2.28)–(2.31) form a closed system,
which should be supplemented only by boundary
conditions. Classic types of boundary conditions
are conditions in terms of stresses or displacements. The temperature distribution is usually
determined from a stationary solution
T;ii ¼ 0
ð2:32Þ
or non-stationary heat conductivity problem with
corresponding boundary conditions.
The system of Eqs. (2.28)–(2.31) may be
converted to a system, similar to the Lamé’sequations. For this purpose, let us express elastic
deformations eEij through the displacement vector
30
2 Deformation and Fracture of Rocks in the Presence of Filtration
ui and temperature DT change by means of
(2.28)–(2.29)
eEij ¼
aT
1
ui;j þ uj;i DTdij
2
3
ð2:33Þ
Then let us substitute the obtained values into
Hooke’s law (2.30)
rij ¼ l ui;j þ uj;i þ kuk;k dij KaT DTdij
ð2:34Þ
2
3l
where K ¼ k þ
is compression modulus. This
relation is referred to as Duhamel-Neumann
equation.
Substitution (2.34) into equation of equilibrium (2.31) gives
l ui;ji þ uj;ii þ kuk;ki dij KaT DT;i dij þ fj ¼ 0
ð2:35Þ
or after a some transformations
ðl þ kÞui;ji þ luj;ii KaT DT;j þ fj ¼ 0 ð2:36Þ
It is seen from here that thermo-elasticity can
be described in terms of Lamé’s equations if
volume forces are formally supplemented by
value of
Dfj ¼ KaT DT;j
ð2:37Þ
The system of Eqs. (2.36), (2.32), together
with the boundary conditions for stresses (or
displacements) and temperatures T are usually
referred to as a system of equations of uncoupled
thermo-elasticity. In that case temperature is
included in Eq. (2.36) as an external variable: its
distribution does not depend on the displacements uj . Thus, the problem may be solved
consequently: starting from finding temperature
distribution from heat conductivity problem
(2.32) follows by solving elasticity problem
(2.36) for the obtained temperature distribution.
The non-stationary analogue of system (2.36),
(2.32), has the form, for example, Novatsky (1975)
u¼0
ðl þ kÞui;ji þ luj;ii KaT DT;j þ fj q€
ð2:38Þ
1
KaT T0
T;ii T_ u_ i;i ¼ 0
j
kT
ð2:39Þ
Here q is density; j is thermal conductivity
coefficient; kT is thermal conductivity coefficient;
T0 is reference temperature; points above the
variables indicate private time derivatives.
The system (2.38), (2.39) becomes coupled:
both temperature T and displacements uj are
included in both equations. Extra terms in these
equations can occur when considering dissipative
processes. If the last term in Eq. (2.39) is
neglected, which is acceptable for solving some
problems, the system ceases to be coupled. Note
that coupling occurs due to presence of terms
with time derivative.
In case of anisotropy, Eq. (2.33) is generalized as
eEij ¼
1
ui;j þ uj;i aTij DT
2
ð2:40Þ
where aTij is where tensor of thermal expansion.
The analogue of Eqs. (2.34) and (2.36) take the
form
rij ¼ Kijkl uk;l Kijkl aTkl T
ð2:41Þ
Kijkl uk;lj Kijkl aTkl T;j þ fj ¼ 0
ð2:42Þ
Analogy for Equations and Boundary
Conditions
Pore-elasticity Eq. (2.16) coincide with the
thermo-elasticity Eq. (2.36), if we set
2.2 Equations of Poroelasticisity
KaT ¼ aP
DT ¼ p
31
ð2:43Þ
There is also a complete analogy for the
equations reflecting the relationship between the
total stresses and total strains (2.18) and (2.34).
For an anisotropic media, instead of the first
Eq. (2.43), we have
Kijkl aTkl ¼ aPij
ð2:44Þ
This analogy makes it possible to use the
solutions of thermo-elasticity problems for
poroelasticity and vice versa. Besides, this analogy allows using software packages containing
thermoplastic modules for solving the problems
of poroelasticity, formally replacing pressure
with temperature and defining the coefficient of
thermal expansion according to (2.43).
However, it should be keep in mind that the
boundary conditions for poroelasticity are usually set for effective stresses sij . When using
ready-made thermo-elastic solutions or application packages, the total stresses must be specified
as the boundary conditions. Their recalculation
from effective through (2.9) does not bring
difficulties.
There are some serious difficulties during
calculation of critical conditions and, especially,
at calculations of inelastic deformation in the
framework of the theory of yield flow. Difficulties are related to the fact that yield functions and
plastic potentials according to the used concept
should be recorded for effective stresses; the
expression of these criteria through the total
stresses by means of (2.21) leads to the appearance of additional parameter p in those functions.
When using the analogy considered, the parameters of the yield criteria and plastic potentials
become formally dependent on the pressure (or
its analogue—temperature).
These difficulties of transition between these
theories arise due to differences in the building of
these theories. Thus, when considering the phenomenon of thermo-elasticity, the strain field was
subjected to decomposition (into elastic and
inelastic parts), and when considering
poro-elasticity, the stress field was subjected to
decomposition (into effective stresses and fluid
pressure). Therefore, while maintaining the
analogy of the form of the final equations, it is
necessary to require full compliance of the total
stresses and total strains for both cases.
We need also to emphasize that the analogy
was considered for uncoupled variants of both
poroelasticity and thermoelasticity. If we consider poroelasticity problem (2.24), the analogy
with the coupled thermoelasticity problem is not
observed: the poroelasticity problem is coupled
already in the static variant, while the thermoelectricity problem becomes coupled only in the
dynamic statement.
2.3
Inelastic Deformation
with Regard to Filtration
Before reaching the yield stress, the mechanical
behavior of the rock is subject to equations of the
poroelasticity considered above. To sum them
up, they can be written for a rather general case
of arbitrary anisotropy as follows
rij;i ¼ 0
ð2:45Þ
sij ¼ rij þ aP pdij
kij p;i ;j ¼ 0
ð2:46Þ
ð2:47Þ
sij ¼ Kijkl eElk
ð2:48Þ
eEij ¼ eTij ¼
1
ui;j þ uj;i
2
ð2:49Þ
Equations (2.45)–(2.49), and together with
the boundary conditions for stresses (or displacements) and pressures, form a closed system.
Let us remind that here rij ; sij are tensors of full
and effective (belonging to the ground skeleton)
stress; eEij ; eTij are tensors of elastic and total strain;
ui is the displacement vector; p is pore pressure;
Kijkl is tensor of elasticity; kij is tensor of permeability; dij is the unit tensor; 0 aP 1 is
Biot’s coefficient that reflects the nature of the
pore space structure; for well-permeable rocks aP
approaches from below to unity. In most cases, it
32
2 Deformation and Fracture of Rocks in the Presence of Filtration
is possible to set aP ¼ 1 for practical
calculations.
In the case of transverse isotropic medium, the
equations of Hooke’s law take the form of (1.15)
with the replacement of full stresses with effective ones
s11 ¼ C11 eE11 þ C12 eE22 þ C13 eE33
s22 ¼ C12 eE11 þ C12 eE22 þ C13 eE33
s33 ¼ C13 eE11 þ C13 eE22 þ C33 eE33
s12 ¼ 2C66 eE12
ð2:50Þ
s13 ¼ 2C44 eE13
s23 ¼
QL ¼ GLð23Þ ðs22 s33 Þ2 þ GLð13Þ ðs11 s33 Þ2 þ GLð12Þ ðs11 s22 Þ2
i1=2
þ 2LLð23Þ s223 þ 2LLð13Þ s231 þ 2LLð12Þ s212
ð2:51Þ
Here, as before, Cij are the coefficients of the
matrix of elasticity for the transverse isotropic
body; k; l are Lamé’s constants.
It follows from the analysis of experimental
data that permeability depends essentially on the
history of the stress-strain state. As an approximation we will consider permeability as a function of the achieved intensity of effective shear
stresses si to be determined from the experiments
k ¼ kðsi Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
sjk sii djk
3
1
sjk sii djk
3
ð2:52Þ
ð2:53Þ
Due to the presence of relation (2.52), the
problems of filtration and deformation become
coupled. In the presence of plastic (inelastic)
strains, Eq. (2.49) should be replaced by the one,
which takes into account the existence of
inelastic strains ePij (1.1)
eTij ¼ eEij þ ePij ¼
þ BLð1Þ s11 þ BLð2Þ s22 þ BLð3Þ s33 AðkÞ ¼ 0
ð2:55Þ
sij ¼ 2leEij þ keEkk dij
si ¼
h
F L ¼ GLð23Þ ðs22 s33 Þ2 þ GLð13Þ ðs11 s33 Þ2 þ GLð12Þ ðs11 s22 Þ2
i1=2
þ 2LLð23Þ s223 þ 2LLð13Þ s231 þ 2LLð12Þ s212
h
2C44 eE23
Similarly, for an isotropic body
3
2
To describe inelastic deformation, we will use
a model similar to that used earlier with the
replacement of full stresses with effective ones.
We will use a variant of the theory of plastic flow
with isotropic hardening. Criterion of transition
to inelasticity (1.45), yield function (1.75) and
plastic potential (1.78) will take the form
1
ui;j þ uj;i
2
ð2:54Þ
LQ
LQ
þ BLQ
ð1Þ s11 þ Bð2Þ s22 þ Bð3Þ s33
ð2:56Þ
For LHS model or, respectively, for CRA
model
F C ¼ GCð23Þ ðs22 s33 Þ2 þ GCð13Þ ðs11 s33 Þ2
þ GCð12Þ ðs11 s22 Þ2
þ 2LCð23Þ s223 þ 2LCð13Þ s231 þ 2LCð12Þ s212
þ BCð1Þ s11 þ BCð2Þ s22 þ BCð3Þ s33 AðkÞ
ð2:57Þ
QC ¼ GCð23Þ ðs22 s33 Þ2 þ GCð13Þ ðs11 s33 Þ2
þ GCð12Þ ðs11 s22 Þ2
þ 2LCð23Þ s223 þ 2LCð13Þ s231 þ 2LCð12Þ s212
CQ
CQ
þ BCQ
ð1Þ s11 þ Bð2Þ s22 þ Bð3Þ s33
ð2:58Þ
If inelastic volumetric deformations are absent
or negligible, there is a restriction (1.80)
LQ
LQ
BLQ
ð1Þ þ Bð2Þ þ Bð2Þ ¼ 0
ð2:59Þ
2.3 Inelastic Deformation with Regard to Filtration
Here, GLðijÞ , LLðijÞ , BLðiÞ , BLQ
ðiÞ —the constants of
the material reflecting its strength anisotropy.
To ensure “deviator associativity” it is necessary (1.82) to set
;
BLQ
ðiÞ ¼ BðiÞ B0 ;
B0 ¼
3
1X
BL
3 j¼1 ðjÞ
ð2:61Þ
@F
@F
dk ¼ 0
dsij þ
@sij
@k
GLð13Þ ¼ GLð23Þ ¼ GLð12Þ ¼ G; LLð13Þ ¼ LLð23Þ
¼ LLð12Þ ¼ 3G; BLð1Þ ¼ BLð2Þ ¼ BLð3Þ ¼ B=3;
ð2:62Þ
(2.55), (2.57) are reduced to Drucker-Prager
criterion (1.26) (Drucker and Prager 1952).
Similar expressions for the constants of plastic
potential (2.58) are obtained by replacing the
index L in the expressions (2.60) by C.
Isotropic hardening is controlled by a change
in parameter AðkÞ, here and after, the argument
k is be concretized as the work of plastic
deformation
dk ¼ sij depij
ð2:63Þ
Under active loading, further growth of
stresses is accompanied by growth of plastic
strains, which is to be described by the law of
plastic flow. For an adequate description of
inelastic deformation, an unassociated law with a
plastic potential (2.56) will be used to ensure
“deviator associativity”
@Q
@rij
ð2:64Þ
ð2:65Þ
Substituting the dk values here gives
@F
@F
sij depij ¼ 0
dsij þ
@sij
@k
ð2:66Þ
Expressing the increase of plastic strain
through the plastic potential (2.64) we obtain
dk ¼
For an isotropic body
dePij ¼ dk
The parameter dk is to be determined from the
condition of the location on the yield surface into
the stress space at an active loading. The condition for the increment of the function F follows
from (2.47)
ð2:60Þ
For a transversally isotropic medium with a
normal isotropy plane the n3 number of parameters in (2.55) to (2.56) is reduced to five by
meeting the conditions (1.46)
GLð13Þ ¼ GLð23Þ ; LLð13Þ ¼ LLð23Þ ; LLð12Þ
¼ GLð13Þ þ 2GLð12Þ ; BLð1Þ ¼ BLð2Þ
33
@F
@Q
dsij =H
sij
@sij
@sij
ð2:67Þ
Here H ¼ Ep1 ¼ @F
@k is a characteristic of
material to be determined from experiments; Ep
has a meaning of the plastic modulus, and in the
first approximation can be considered as a constant. Functions F; Q should be considered as
F L ; QL or F C ; QC .
Thus, we have system of differential
Eqs. (2.64), (2.63), (2.67) for the given F; Q; H.
References
Biot MA (1935) Le problème de la consolidation des
matières argileuses sous une charge. Ann Soc Sc de
Brux Ser. B. 55:110–113
Biot MA (1941) General theory of three-dimensional
consolidation. J Appl Phys 12:155–156
Darcy H (1856) Les Fontaines Publiques de la Ville de
Dijon, Dalmont, Paris
De Witt R (1970) Linear theory of static dislocations. Nat
Bur Stand US 1:651
De Witt R (1973) Theory of disclinations II continuous
and discrete disclinations in anisotropic elasticity.
J Res Natn Bur Stand 77A:49
Drucker DC, Prager W (1952) Soil mechanics and plastic
analysis for limit design. Quart Appl Math 10(2):157–
165
Duhamel JMC (1837) Second mémoire sur les phénomènes thermo-mécaniques. J de l’École Polytechnique
15(25):1–57
34
2 Deformation and Fracture of Rocks in the Presence of Filtration
Duhamel JMC (1838) Mémoire sur le calcul des actions
moléculaire développées par les changements de
tempé rature dans les corps solides. Mémoires
présentées par diver savant à l’Ac ad. des sciences
5:440–498
Khristianovich SA, Zheltov YuP (1955) About the
hydraulic fracturing of the oil-bearing formation.
Izv USSR Acad Sci 5:3–41
Lekhnitsky SG (1977) Anisotropic body elasticity theory.
M.: Science. 415p
Neumann F (1885) Vorlesung über die Theorie des
Elasticität der festen Körper und des Lichtäthers.
Teubner, Leipzig
Novatsky V (1975) The theory of elasticity. M: Peace.
256p
Terzagi K (1925) Erdbaumechanik auf Bodenphysikalischen Grundlagen. Deuticke, Wien
3
Mechanical and Mathematical,
and Experimental Modeling of Oil
and Gas Well Stability
Nowadays, technologies of oil and gas field
development based on drilling of inclined and
horizontal wells, as well as underbalanced drilling when drilling mud pressure in the well
below the oil (gas) formation pressure, are
becoming more and more widespread. However,
there are problems associated with wellbore
instability, which did not exist before.
For the first time in Russia, the problem of
loss of stability of rocks that compose walls of
drilling wells has arisen in a number of oil fields
in the south of the country, where rock collapses
occurred during the development of deep-lying
horizons. There are many cases of failures of oil
and gas wells, which have opened salt and clay
rocks. The destruction of wellbore was observed
on a number of oil fields of Western Siberia, as
for individual wells, as well as for whole well
clusters.
As a result of studies on the causes of rock
collapse, various hypotheses explaining these
causes appeared. For a long time, the main reason for the collapses in wells was considered the
swelling of clays composing the walls of the
well, due to the absorption of water from the
drilling mud (Rzhanitsyn and Tsarevich 1936).
In the work of Dinnik (1925), when studying the
issue of borehole stability, the state of rocks in
the well vicinity is considered in the process of
drilling, because it is the formation of a rock
opening accompanied by the volume uneven
compression of surrounding rocks and the
physical and chemical impact of the fluid on
© Springer Nature Switzerland AG 2020
V. Karev et al., Geomechanics of Oil and Gas Wells,
Advances in Oil and Gas Exploration & Production,
https://doi.org/10.1007/978-3-030-26608-0_3
them, entails instability of rocks. For the case
when the well is filled with liquid that creates
back pressure on the walls, Lekhnitsky (1977)
proposed formulas to determine the three principal normal stresses.
Analyzing the reasons of complications during well drilling in Bashkiria, Isaev (1958) came
to the conclusion that the main reason for rock
failure during drilling is the rock pressure, and
the role of hydration and swelling in the interaction with flushing fluids is reduced to a change
in the mechanical properties of rocks and, consequently, to a decrease or increase in the degree
of manifestation of rock pressure. Vasiliev and
Dubinina (2000) called the following reasons of
rock failure in the bottom-hole zone: redistribution of stresses caused by the weight of overlying
rocks and the reduction of formation pressure;
filtration of liquid or gas to the wellbore. The
work assesses the stresses caused by the above
causes.
Petukhov and Zapryagaev (1984) experimentally studied deformations of the walls of
uncased wells of various diameters depending on
the type of stress state of the mountain massif.
They results are proposed to be used to determine
the stability of the walls of uncased wells at
various depths by solving the issue of rock
strength, taking into account the coefficient of
structural weakening, temperature factor and
consistency of drilling mud.
Blokhin and Terentyev (1984) proposed a
method for calculating the size and nature of the
35
36
3
Mechanical and Mathematical, and Experimental Modeling …
distribution of normal stresses and displacements
in the bottom-hole zone of well. The method was
developed using the measured in situ hydraulic
fracturing pressures. The work of Katsaurov
(1972) presents a formula for determining the
radius of the inelastic deformation zone taking
into account cohesion of rocks in this region.
Wellbore damage during drilling can occur by
various mechanisms (Spivak and Popov 1994).
The wellbore stability is affected by various
factors, the main of which are the ratio between
the inclination of the well, the amount of inclination of the formation, the difference in strength
properties of the rock in the direction along the
layering and the normal to them (Aoki et al.
1994). This is due to the fact that drilling and
operation of wells affect the local stress-strain
state in the formation. As the stresses on the
wellbore walls are redistributed, under certain
conditions, the shear stresses may exceed the
rock ultimate strength, which leads to the
destruction of rock and loss of wellbore stability
during drilling and sand production during
operation. The nature of well damage will
depend on the mechanical properties of the
material, as well as the distribution of initial
stresses in formation.
From the analysis of a question condition for
today it is possible to draw a conclusion that
methodical workings out on strength calculations
of a wellbore are executed now mainly for vertical wellbores. At the same time, the main tool
of oil and gas field development is gradually
becoming the drilling of inclined and horizontal
wells, including underbalanced drilling. The
peculiarity of such stability problems lies in the
fact that the anisotropy of the deformation and
strength properties of rocks in which a well is
drilled comes to the fore. n addition, the inclined
sections of the well are fundamentally different
from the vertical ones in that the stress state of
the rocks adjacent to them is not asymmetrically
relative to the axis of the well. Today these
questions are studied insufficiently and require a
comprehensive study.
The solution to any geomechanical problem
involves answering two questions. The first is the
stresses that occur in the rock during certain
operations in the formation. The second is the
reaction of rocks in terms of their deformation
and destruction to these stresses.
The answer to the first question does not have
any fundamental difficulties, since the numerical
methods to solve it are well developed. To calculate the stresses, it is necessary to know the
elastic and strength characteristics of rocks under
study. For anisotropic rocks, this either requires
the use of true triaxial test facilities, such as the
TILTS installation, or a series of indirect measurements by standard test systems on a number
of specimens cut at different angles, followed by
a recalculation of parameters.
The answer to the second question is usually a
little more difficult. This is due to the fact that, as
indicated above, attempts to create an adequate
mechanical and mathematical model describing
the processes of rock destruction in the vicinity
of an inclined well, taking into account changes
in the angle of its inclination, for highly anisotropic rocks lead to a sharp complication.
The Institute for Problems in Mechanics of the
Russian Academy of Sciences has developed a
fundamentally new approach to solving problems
of wellbore stability. It is based on the physical
modeling of deformation and fracture processes
in the vicinity of inclined and horizontal wells by
using the unique Triaxial Independent Load Test
system TILTS created at the Institute. The
facilities, as noted in Chap. 4, allows loading of
cubic rock specimens independently in three
directions. This opens up the possibility to fully
reproduce in the laboratory the real stress states
arising in the vicinity of oil and gas wells during
their drilling, completion and operation, and to
study the influence of stress on the processes of
deformation and destruction in these areas.
3.1
Stress State in the Vicinity
of the Well in Isotropic Rocks
One of the key challenges that must be addressed
before experiments can be carried out is the
development of loading programs for specimens
when testing them by using TILTS. Rock specimens should be being loaded according to the
3.1 Stress State in the Vicinity of the Well in Isotropic Rocks
loading programs corresponding to the stresses
that occur in the vicinity of the well during its
drilling.
As is known, tangential (shear) stresses lead
to the destruction of materials. Pressure drawdown, which is a decrease in bottom-hole pressure compared to reservoir pressure, leads to
changes in stress-stain state around the well. The
increase of pressure drawdown results in growth
of shear stresses in the vicinity of the well, which
may eventually lead to rock destruction (cracking, loosening). The changes of stress-strain state
near the well with the pressure drawdown
increase for various options of bottom-hole
design were studied by the help of mathematical modeling: analytical modeling for simple
cases (open hole), numerical modeling using
three-dimensional programs in more complex
cases (the presence of casing, perforation holes,
slots, etc.). Thus, the changes of stresses distribution with the drawdown increase were determined, i.e., loading programs of the specimens
for each variant of the bottom-hole design: open
borehole, cased borehole, perforation, horizontal
or vertical slots on the well wall. The tests allow
determining the stresses (value of pressure
drawdown) corresponding to beginning of
inelastic deformation accompanied by increase or
decrease in permeability. Visual observation of
the specimens after testing reveals the type of
inelastic deformation: cracking, loosening, plastic yield, etc.
There are a number of practically important
cases when it is possible to obtain analytical
solutions for stresses in the vicinity of the well,
and in this case the problem of building loading
programs is greatly simplified. First of all, this
applies to a situation where:
(a) The rock in which the borehole is drilled is
isotropic in both deformation and strength
properties;
(b) The natural stress state can be considered
close to the state of uniform hydrostatic
compression by the rock pressure at a given
depth. This can primarily be expected in the
absence of significant geological disturbances for rocks composing the formation
37
being sufficiently plastic, so that during
geological times all stresses in the formation
had to be leveled out.
However, even in the case of an uneven initial
stress state, in particular in the presence of lateral
rock pressure different from the vertical pressure,
it is possible to develop loading programs for a
number of practically important cases, in particular for horizontal wells drilled along the direction of maximum and minimum horizontal
principal stresses. For this purpose solutions of
two classical problems of elasticity theory,
Lamé’s problem and Kirsch’s problem, should be
used.
Lamé’s Problem and Kirsch’s Problem
Lamé’s problem devoted to the stress state in a
thick-walled hollow cylinder loaded with uniform internal and external pressure, constant
along the length of the pipe. Consider a
thick-walled cylinder with internal a and external
b radii, subjected to the action of uniformly
distributed internal pa and external pb pressures,
respectively, Fig. 3.1 (Timoshenko and Goodier
1979). In cylindrical coordinate system r; h; z the
general solution of Lamé’s equation for radial rr
and circumferential rh stresses is
Fig. 3.1 Lamé’s problem. Configuration, applied loads
38
3
A
þ 2C
r2
A
rh ¼ 2 þ 2C
r
rr ¼
Mechanical and Mathematical, and Experimental Modeling …
rr ¼ 0
ð3:1Þ
Here A and C are the integration constants
determined from the conditions on the inner and
outer surfaces of the cylinder, where pressures,
i.e. normal stresses rr , are known:
ðrr Þr¼a ¼ pa
and ðrr Þr¼b ¼ pb
ð3:2Þ
Here, as everywhere else, it is accepted that
the compressive stresses are negative.
Then, by substituting (3.2) into the first of
Eq. (3.1) and determining constants A and C we
obtain formulas for stresses rr and rh
a2 pa b2 pb ðpa pb Þa2 b2
2 2
b2 a2
r ðb a2 Þ
a2 pa b2 pb ðpa pb Þa2 b2
rh ¼
þ 2 2
b2 a2
r ðb a2 Þ
rr ¼
ð3:3Þ
The Kirsch’s problem is the problem of uniaxial stretching of a plate with a circular hole,
Fig. 3.2. Here a polar coordinate system r; /
with the origin at the hole center is introduced,
related to the Cartesian system as follows:
x ¼ r cos /;
y ¼ r sin /;
x2 þ y2 ¼ r 2
/ ¼ arctg
y
;
x
ð3:4Þ
The stresses in the plate with a circular hole of
radius a stretched along x-axis of the Cartesian
coordinate system (Fig. 3.2) are (Timoshenko
1937)
S
a2
S
1 2 þ
1þ
2
2
r
S
a2
S
1þ 2 1þ
r/ ¼
2
2
r
rr ¼
3a4 4a2
cos 2/
r4
r2
3a4
cos 2/
r4
ð3:5Þ
It follows from (3.5) that at the points of the
contour of the hole at r ¼ a
r/ ¼ Sð1 2 cos 2/Þ
ð3:6Þ
This means that the normal circumferential
stresses r/ is maximal at / ¼ 90 , i.e., the points
M of the hole contour lying on the axis
¼ 3S; for points N lying on
y (Fig. 3.2), and rmax
/
the axis x, r/ ¼ S. It was noted that one of the
simplest but at the same time one of the most
important for practice modeling tasks is modeling the stress-strain state near an uncased well
drilled in isotropic rocks, subjected to uniform
hydrostatic compression. In this case, the stresses
acting in the vicinity of the well do not depend
on its angle of inclination from the vertical.
Therefore, loading programs for the TILTS
simulation will be the same for vertical, horizontal and inclined wells.
Stress State in the Vicinity of Uncased
Wells
In the initial state, oil and gas reservoirs are
usually subjected to uniform compression by
rock pressure. In the absence of pronounced
geological disturbances, the vertical stress is
determined by the weight of the overlying rocks.
The lateral rock pressure may generally differ
from the vertical one. However, if the rock surrounding the formation is sufficiently plastic,
then during geological times all shear stresses in
the formation had to be relaxed, so that we can
assume that the rock pressure in the undisturbed
formation is the same in all directions, i.e. each
element of the rock is evenly compressed from
all sides.
In accordance with this, the stress state of the
formation in the initial state will be considered as
a state of uniform hydrostatic compression by the
rock pressure q ¼ ch, where c is the average
specific weight of the overlying rocks; h is the
depth.
Two cases should be distinguished: permeable
and impermeable rocks, corresponding to oil and
gas reservoirs, and the surrounding rocks,
respectively. The effective (transmitting by rock
skeleton) stresses differ from the total stresses by
3.1 Stress State in the Vicinity of the Well in Isotropic Rocks
39
Fig. 3.2 Stretched plate with a circular hole. Kirsch’s problem
the value of pore pressure for the former case and
coincide with the total stresses for the latter case.
Permeable Rock
On Fig. 3.3 the section of the vertical well and
the stresses acting in its vicinity are shown.
The rocks forming the reservoirs of oil and
gas fields (sandstones and limestone) possess
mainly a granular structure. The stresses acting
rr ; r/ ; rz in the vicinity of the well are partially
taken by fluid pressure p and partially by the
stresses sr ; s/ ; sz transmitted through the contacts
between the grains of the rock (effective stresses)
(2.20)
rr ¼ sr aP p;
r/ ¼ s/ aP p;
rz ¼ sz aP p
ð3:7Þ
For most permeable rocks aP ¼ 1, so in the
future, for simplicity we will assume
r/ ¼ s/ p;
r0r
¼
r0/
¼
r0z
rr ¼ sr p;
¼ q;
s0r
¼
s0/
r z ¼ sz p
¼ s0z ¼ q þ p0
ð3:8Þ
Here p0 is the initial fluid reservoir pressure.
For isotropic media and equip component
rock pressure q, the distribution of the total
stresses caused by the action of rock pressure in
the vicinity of the well does not depend on the
angle of inclination and is determined by the
known solution of the Lamé’s problem (3.3).
Assuming in (3.3), a ¼ Rw , a=b ¼ 0, ðrr Þr¼b ¼ q,
ðrr Þr¼Rw ¼ pw , we get
rr ¼ ðq þ pw ÞðRw =r Þ2 þ q
r/ ¼ ðq þ pw ÞðRw =r Þ2 þ q
rz ¼ q
Fig. 3.3 Stresses acting in the vicinity of a vertical well
ð3:9Þ
40
3
Mechanical and Mathematical, and Experimental Modeling …
Here q is rock pressure (q < 0), pw is pressure
in the well ðpw [ 0Þ; Rw is the well radius; r is
the distance from the well axis.
Shear stresses s ¼ 1=2 rr r/ are equal to
s ¼ ðq þ pw ÞðRw =r Þ2
ð3:10Þ
On the borehole wall, i.e. at r ¼ Rw , from
(3.9) we have
rr ¼ pw ;
r/ ¼ 2q þ pw ;
rz ¼ q ð3:11Þ
Then from (3.9), and (3.8), the value of
effective stresses in the vicinity of the well, are
determined by
sr ¼ ðq þ pw ÞðRw =r Þ2 þ q þ pðrÞ
s/ ¼ ðq þ pw ÞðRw =r Þ2 þ q þ pðrÞ
sz ¼ q þ pðrÞ
ð3:12Þ
where pðrÞ is pressure at a distance r from the
well contour.
It follows from (3.12) that on the wellbore
contour, the effective stresses are
sr ¼ 0
s/ ¼ 2ðq þ pw Þ
sz ¼ q þ pw
s s /
r
s¼
¼ j q þ pw j
2
ð3:13Þ
sr ¼ ðq þ pw ÞðRw =rÞ2 þ q
ð3:14Þ
sz ¼ q
On the wellbore contour, i.e. at r ¼ Rw , the
stresses are
sr ¼ pw
s/ ¼ 2q þ pw
sz ¼ q
s ¼ jq þ pw j
qb3 ðr 3 a3 Þ pw a3 ðb3 r 3 Þ
3 3
r 3 ð b3 a3 Þ
r ð b a3 Þ
3
3
qb ð2r þ a3 Þ pw a3 ð2r 3 þ b3 Þ
þ
rh ¼ ru ¼ 3 3
2r ðb a3 Þ
2r 3 ðb3 a3 Þ
ð3:16Þ
rr ¼ Impermeable Rock
In impermeable layers, the effective stresses are
equal to the total stresses, i.e. sij ¼ rij .
s/ ¼ ðq þ pw ÞðRw =rÞ2 þ q
Stress in the Vicinity of the Perforated
Hole
The vast majority of productive wells are cased.
Therefore, it is important to obtain loading programs to simulate the stress states corresponding
to that occur near the perforation holes.
As before, we will consider the formation as
isotropic in its deformation properties, and the
rock pressure to be hydrostatic.
The stresses in the vicinity of the perforation
hole vary along its length. Two zones can be
distinguished: near the walls of the perforation
hole and near its tip.
The stress state near the walls of the perforation
hole far from both its end and the borehole contour
can be accurately approximated by expressions
(3.12) and (3.13) for an infinite open hole. The
stresses acting in the vicinity of the tip of the
perforation hole can be approximated by the
stresses acting in the vicinity of the spherical cavity,
Fig. 3.4.
The distribution of stresses in the hollow
sphere with internal a and external b radii,
symmetrically loaded by internal pw and external
q pressures, respectively, in the spherical
coordinate system is (Timoshenko and Goodier
1979)
Here rr , rh , r/ are radial and two circumferential stresses (Fig. 3.4).
Then, specifying the values in (3.17) to be
corresponded to a perforation hole:
a ¼ Rw ;
ðrr Þr¼Rw ¼ pw ;
at
a=b ¼ 0;
ðrr Þr¼b ¼ q
ð3:17Þ
ð3:15Þ
Expressions (3.16) are reduced for total
stresses to the following expressions
3.1 Stress State in the Vicinity of the Well in Isotropic Rocks
41
Fig. 3.4 Stresses in the vicinity of the spherical cavity
rr ¼ ðq þ pw ÞðRw =r Þ3 þ q
rh ¼ 1=2ðq þ pw ÞðRw =r Þ3 þ q
ð3:18Þ
sr ¼ 0
sh ¼ 3=2ðq þ Pw Þ
3
r/ ¼ 1=2ðq þ pw ÞðRw =r Þ þ q
On the wall of the perforation hole (at r ¼ Rw )
rr ¼ pw ;
rh ¼ 3=2q þ 1=2pw ;
r/ ¼ 3=2q þ 1=2pw
ð3:21Þ
s/ ¼ 3=2ðq þ Pw Þ
ð3:19Þ
For permeable rocks, in which perforation is
mainly used, the effective stresses in the vicinity
of the tip of the perforation hole is obtained from
(3.18) and (3.8)
sr ¼ ðq þ pw ÞðRw =r Þ3 þ q þ pðrÞ
sh ¼ 1=2ðq þ pw ÞðRw =r Þ3 þ q þ pðrÞ
On the borehole wall, i.e. at r ¼ Rw , from
(3.20) we have
ð3:20Þ
s/ ¼ 1=2ðq þ pw ÞðRw =r Þ3 þ q þ pðrÞ
where pðrÞ is pressure at a distance r from the
well.
Stresses in the Vicinity of the Well in Isotropic Formation Under Uneven Initial
Stress State
The initial stress state of the formation may
diverge from the considered hydrostatic one. In
the general case it is determined by the weight of
the overlying rocks, the geological structure of
the massif, tectonic processes and is characterized by three principle stresses and their orientation in space (e.g. in terms of three Euler’s
angles). The problem of measuring the complete
stress tensor in situ attract a lot of forces of
researches but is still far from solving. Therefore,
42
3
Mechanical and Mathematical, and Experimental Modeling …
it is usually supposed that one of the principle
stresses is aligned vertically and is determined by
the weight of overlying rocks, qV ¼ ch, where
c is the average specific weight of the overlying
rocks, h is the depth. Two other principal stresses
are directed along two orthogonal axes lying in
the horizontal plane and characterized by the
maximum and minimum values of horizontal
min
rock pressure qmax
and qmin
(qmax
H
H \0, qH \0
max Hmin and qH [ qH ) (Zobak 2007; Goodman
1980; Jaeger et al. 2007).
If the reservoir through which the well is
carried out is permeable, the ground skeleton far
from the well is loaded with effective stresses:
vertically ðqV þ p0 Þ, horizontally ðqmax
H þ p0 Þ and
ðqmin
þ
p
Þ,
where
p
[
0
is
the
oil or gas
0
0
H
reservoir pressure. The pressure inside the well is
equal to pw [ 0.
The question of the influence of the natural
stress state on the wellbore stability has recently
acquired a special attention in relation with the
technology of oil and gas production with the
help of horizontal wells. This factor is important
also for vertical wells, because deviation of the
initial stress field from hydrostatic pressure has a
significant impact on the distribution of stresses
on the vertical well contour, resulting in formation of cracks, rock falls from the walls of the
well, the and etc.
The deviation of the initial stress field from
hydrostatic pressure results in directional
dependence of stress distribution in the vicinity
of well, and also in dependence of the stress
distribution (and as a sequence of the wellbore
stability) on the orientation of the well relative to
directions of the principle stresses.
Physical modeling of real processes of rock
deformation and destruction in the vicinity of
horizontal wells under non-equicomponent stress
field can be performed using the experimental
installation of the Institute of Mechanics Problems of the Russian Academy of Sciences—
Testing system of three-axis independent loading
(TILTS). To perform these tests it is necessary to
develop loading programs that meet the actual
stress conditions occurring in the vicinity of
wells, which requires to know the change of
stresses in the vicinity of the well during the
decrease of the bottom-hole pressure.
Depending on the values of the initial stresses,
there will also be different stresses arising in the
vicinity of the wells. The values of these stresses
also depend on the mutual orientation of the
principle stresses and the drilling direction of the
well. Therefore, the loading programs for the
physical simulation of drilling and well operation
on the TILTS should be chosen accordingly. In
general, for their development it is necessary to
carry out rather complex three-dimensional calculations to determine the stresses acting in the
vicinity of the well. However if the well direction
coincides with the one of directions of the principle stresses, the loading programs can be
obtained with the help of analytical solutions.
Vertical Well
Let find the stresses acting on the vertical well
contour at the unequal natural stress state of the
reservoir. Figure 3.5 represents the horizontal
cross-section of the vertical well and the horizontal stresses acting in the reservoir away from
the well. The problem under consideration can be
represented as a superposition of two problems:
Fig. 3.5 Horizontal section of a vertical well and natural
stresses acting in the reservoir
3.1 Stress State in the Vicinity of the Well in Isotropic Rocks
43
For the second problem it follows from the
solution of the Kirsch’s problem circumferential
stresses r/ change along the well contour. They
are minimal and of the opposite sign at point M
(and at the opposite point) and reach a maximum
at point N (and at the opposite point).
At point M, the stresses are:
rr ¼ 0
Fig. 3.6 Lamé’s problem
min
r/ ¼ ðqmax
H qH Þ
rz ¼ 0
ð3:23Þ
At point N, the stresses are:
rr ¼ 0
min
r/ ¼ 3ðqmax
H qH Þ
rz ¼ 0
Fig. 3.7 Kirsch’s problem
1. The hydrostatic compression with stresses
qmin
H applied far away from the well contour.
The pressure pw applied at the well contour,
Fig. 3.6. This problem is known as the Lamé’s
problem and is considered above in p. 3.1.
2. Uniaxial compression in the direction of the
maximum horizontal stress action ðqmax
H qmin
Þ
applied
far
away
from
the
well,
Fig.
3.7,
H
with no pressure applied inside the well. This
problem is known as the Kirsch’s problem
and is also discussed in p. 3.1 above.
For the first problem it is followed from the
solution of Lamé’s problem that the radial, circumferential and axial stresses in all points on the
vertical well contour will be identical and equal to
r/ ¼
þ pw
rz ¼ qV pw
The total stresses acting on the well contour
are equal to the sum of the stresses for each of the
specified problems.
For the total stresses at point M by summing
(3.22) and (3.23) we find:
rr ¼ pw
max
r/ ¼ 3qmin
H qH þ pw
ð3:25Þ
rz ¼ qV
Similarly, for the total stresses at point N by
summing (3.22) and (3.24) are equal:
rr ¼ pw
min
r/ ¼ 3qmax
H qH þ pw
rz ¼ qV
ð3:26Þ
Accordingly, for the permeable layer, the
effective stresses acting in the soil skeleton ðSi ¼
ri þ pw Þ are equal to:
sr ¼ 0
rr ¼ pw
2qmin
H
ð3:24Þ
ð3:22Þ
max
rh ¼ 3qmin
H qH þ 2pw
sz ¼ qV þ pw
ð3:27Þ
44
3
Mechanical and Mathematical, and Experimental Modeling …
for point M
sr ¼ 0
min
r/ ¼ 3qmax
H qH þ 2pw
s z ¼ qV þ pw
ð3:28Þ
for point N, respectively.
Horizontal Well
Let us find the stresses acting on the horizontal
well contour at an uneven natural stress state of
the formation, Fig. 3.8.
Contrary to the case of the vertical well, the
distribution of stresses on a horizontal well
contour will depend on its direction relative to
the directions of the maximum and minimum
horizontal stresses.
Two cases will be considered below:
– the axis of the
the direction
compression;
– the axis of the
the direction
compression.
horizontal well coincides with
of the maximum horizontal
horizontal well coincides with
of the minimum horizontal
For these two cases, the expressions for the
stresses on the well contour can be obtained
analytically.
Fig. 3.9 A horizontal well drilled along the maximum
horizontal stress
The Well Drilled Along the Direction of the
Maximum Horizontal Stress
The vertical section of a horizontal well drilled
along the maximum horizontal stress, and the
initial stresses acting in the formation away from
the well are shown on Fig. 3.9. The third principle stress qmax
H acts along the z-axis of the well.
The solution of this problem is obtained from
the solution for the vertical well by formal
replacing qmax
with qV . Then for the points M
H
and N from (3.25) to (3.28) for full stresses ri
and effective stresses si (for the case of a permeable layer) we have:
at the point M
rr ¼ pw
r/ ¼ 3qmin
H qV þ pw
ð3:29Þ
rz ¼ qmax
H
sr ¼ 0
s/ ¼ 3qmin
H qV þ 2pw
sz ¼ qmax
H þ pw
ð3:30Þ
at the point N
rr ¼ pw
rh ¼ 3qV qmin
H þ pw
rz ¼ qmax
H
ð3:31Þ
sr ¼ 0
Fig. 3.8 Stresses acting in the vicinity of a horizontal
well
s/ ¼ 3qV qmin
H þ 2pw
sz ¼
qmax
H
þ pw
ð3:32Þ
3.1 Stress State in the Vicinity of the Well in Isotropic Rocks
The Well Aligned Along the Minimum Horizontal Stress
The vertical section of a horizontal well drilled
along the minimum horizontal stress, and the
initial stresses acting in the formation away from
the well are shown on Fig. 3.10. The third
principle stress qmin
H acts along the z-axis of the
well.
The solution of this problem is obtained from
the solution for the horizontal well by formal
max
replacing qmin
H with qH .
Thus, we have from (3.29) to (3.32) for point M
rr ¼ pw
r/ ¼ 3qmax
H qV þ pw
rz ¼
ð3:33Þ
qmin
H
sr ¼ 0
s/ ¼ 3qmax
H qV þ 2pw
sz ¼
qmin
H
ð3:34Þ
þ pw
and for point N
rr ¼ pw
r/ ¼ 3qV qmax
H þ pw
rz ¼
sr ¼ 0
s/ ¼ 3qV qmax
H þ 2pw
sz ¼
qmin
H
ð3:35Þ
qmin
H
ð3:36Þ
þ pw
Fig. 3.10 A horizontal well drilled along the minimum
horizontal stress
3.2
45
Mechanical Model of Stability
of Inclined and Horizontal Wells
in Anisotropic (Layered)
Formations
It is known from practice of drilling that when
the wellbores reach a certain angle of inclination
(for various rocks, it lies within the range of 40°–
60°) fracture of the wellbores walls is observed
in various forms, which leads to a stop of drilling. Two points should be mentioned:
– The loss of stability of inclined wellbores is
often observed in rocks with pronounced
layering;
– During drilling vertical wells in the same
formations, wellbore wall failure is observed
much less often and at significantly lower
drilling mud densities.
These facts leads to conclusion, that anisotropy of the strength properties of rocks, determined by the presence of planes of weakening, is
an essential factor influencing the stability of
boreholes.
Similar problems arise when operating horizontal wells.
Note, that in the same formations in case of
vertical wells, even under significantly greater
drops down of pressure, wellbore failures do not
observed.
Therefore, question of determining the maximum safe drops down during operation of horizontal and inclined wells arize.
Stress State in the Vicinity of Inclined Wells
Drilled in Layered Rock Massif
The initial stress state is supposed to be
equi-component compression q.
It is known that for a vertical wellbore in a
transversally isotropic rocks, when axis of borehole coincides with the axis of isotropy, the
stress distribution in its vicinity will be the same
as for a wellbore in isotropic medium, and is
given by Lamé’s solution (3.3).
46
3
Mechanical and Mathematical, and Experimental Modeling …
However, if the well is inclined, the situation
changes. The stresses along the well contour are
no longer constant, as in the case of isotropic
media, but varying along the contour, and the
variation depends on elastic constants and the
wellbore inclination. Shear stresses appear in the
planes of weakening, which increase with the
well inclination. Correspondingly, the probability
of rock fracture in these planes and the risk of
well failure increase.
Thus, in general, computation of stresses
acting in the vicinity of an inclined well drilled in
transversally isotropic rock is a complex problem
and requires knowing the values of five elastic
constants for transversally isotropic rock.
However, for most rocks, the problem appears
much easier. As mentioned in Sect. 3.2, the
solution for isotropic medium can be used in
most practical cases for wellbores in transversally isotropic rocks.
Therefore, the solutions for a well in an elastic
isotropic medium (Sect. 3.1) may serve as a good
approximation for determining the stresses in the
vicinity of an inclined well drilled in a
transversally isotropic formation.
Although the distribution of stresses in the
coordinate frame connected to the wellbore axis
is considered as coinciding with the stress
distribution in isotropic medium, the shear and
normal stresses acting on the planes of weakening will depend on the angle of inclination of
the wellbore. The fracture is expected to begin
along these planes, because the strength ½s in
these planes is much lower than in other
directions.
According to (1.37), ultimate effective shear
stresses acting in the rock skeleton planes of
weakening are
½s ¼ sc sn tg qc
critical shear stresses will cover the increasing
sector of the contour.
If the inclination angle continues to increase,
the rock near the well can no longer withstand
the stresses and disintegrates. Such ultimate state,
corresponding stresses and inclination angle of
the well will be referred to as the limiting state.
Thus, reaching the critical state at one or few
points on the well contour is not sufficient for the
wellbore walls to fail. For failure it’s essential
that the destruction cover a sufficiently large area.
Consider a rock mass as a continuous medium, which behaves as isotropic when deformed,
however, the fracture of which may occur along
the weakening planes coinciding with the layering. Then the stress state along the well contour
will not depend on the position of the considered
point on the well contour. However, the presence
of planes of weakening makes points of the well
contour unequal in terms of potential failure.
Hence a problem appears of a choice of the most
dangerous points (or area) on a contour of a well,
i.e. the points, at which the limiting state (3.56) is
reached first.
Figure 3.11 depicts a section of an inclined
well drilled in rock with a horizontal layering.
Here z is the vertical axis, axis; z0 is the well
axis, h is the angle of inclination of the well from
the vertical. The cross-section of the borehole by
the horizontal plane (formation plane) is an
ellipse; u—the angle between the large half-axis
of such an ellipse and the point in question.
The following notation will be used:
ð3:56Þ
where sc ; qc are the adhesion and friction angle
for the planes of weakening.
This means that fracture will begin primarily
at those points in the borehole contour where the
shear stress within the planes of layering reaches
a value ½s. As the calculations show (see below),
as the inclination angle of the well increases, the
Fig. 3.11 Position of potentially dangerous points on the
well contour for small inclination angles
3.2 Mechanical Model of Stability of Inclined and Horizontal Wells …
p0 [ 0 is reservoir pressure; Dp ¼ p0 pw is
bottom-hole pressure drawdown; x; y; z are components of Cartesian coordinate system connected
to formation geometry (z-axis is assumed to be
vertical, normal to the layering); h is the angle of
inclination of the well to the vertical, the axis of
the well is assumed to be in the plane xz; r; u; z0 is
a cylindrical coordinate system connected with
the well; angle u is calculated from the axis x; s is
absolute value of shear stress in the plane of
weakening (horizontal plane); sn is effective stress
normal to the plane of weakening.
To derive a fracture criterion, it is necessary to
calculate the shear stresses in the planes of
weakening and the stresses normal to it. Two
cases will be considered as usual: permeable and
impenetrable rocks.
The first case: permeable rock. The effective
stress state acting in the rock skeleton on the well
contour in this case according to (3.13) is
sz ¼ ðq þ pw Þ ¼ ðq þ p0 DpÞ
sr ¼ 0
ð3:57Þ
The absolute value of shear stresses in the
plane of weakening (horizontal plane) can be
calculated by transforming the components of the
stress tensor to the coordinate system associated
with layering as follows
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 cos2 2u sin2 h
ð3:58Þ
Compressive stresses normal to the plane of
expected attenuation will be
sn ¼ ðq þ p0 DpÞ 1 cos 2u sin2 h ð3:59Þ
The second case; impermeable rock. The
stress state on the well contour according to
(3.11) and for impermeable rocks is
sz ¼ q
sr ¼ pw
su ¼ 2q þ pw
The absolute value of shear stresses in the
plane of weakening (horizontal plane) is
s ¼ ðq þ pw Þ sin h
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 cos2 2u sin2 h
ð3:61Þ
Compressive stresses normal to the plane of
weakening is
sn ¼ q ðq þ pw Þ cos 2u sin2 h
ð3:62Þ
Summing up both cases together, the expressions for shear and compressive stresses for the
plane of weakening may be written as follows
s ¼ B sin h
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 cos2 2u sin2 h
sn ¼ A B cos 2u sin2 h
ð3:63Þ
ð3:64Þ
where for permeable rocks
A ¼ B ¼ q þ pw ¼ q þ p0 Dp
ð3:65Þ
for impermeable rocks
su ¼ 2ðq þ pw Þ ¼ 2ðq þ p0 DpÞ
s ¼ ðq þ p0 DpÞ sin h
47
ð3:60Þ
A¼q
B ¼ q þ pw
ð3:66Þ
Therefore, the most dangerous points on the
contour will be those points for which the condition s ¼ ½s, where ½s is determined by the
ratio (3.56) is satisfied first.
At these points, function
Y ðh; uÞ ¼ s sn tgqc
ð3:67Þ
where s; sn defined (3.63) and (3.64) will have
maximum. Locations of points of the local
maximums of (3.67) are determined from the
condition of equality to zero of its derivative over
u:
@ Y ðh; uÞ
@u
!
sin h cos 2u
¼ sin 2u tgqc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0
1 cos2 2u sin2 h
ð3:68Þ
48
3
Mechanical and Mathematical, and Experimental Modeling …
ð3:69Þ
On finding the values of critical angles u, the
fracture condition is obtained by substitution of
the found values into (3.67). Finally, the value of
the critical angle and fracture condition are given
by the following formulas:
ð3:70Þ
for
Equation (3.68) is satisfied if either
sin 2u ¼ 0
or
sin h cos 2u
tgqc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0
1 cos2 2u sin2 h
Without loosing the generality consider
Eq. (3.70) for 0 qc p=2; 0 h p=2. The
solution of Eq. (3.69) always exists, while the
actual solution of Eq. (3.70) for physically possible values of parameters does not always exist.
Transform (3.73), as follows
cos2 2u sin2 h
1 cos2 2u sin2 h ¼
tg2 qc
1
1 ¼ cos2 2u sin2 h 1 þ 2
tg qc
sin2 h
1 ¼ cos 2u 2
sin qc
ð3:71Þ
sin2 qc
sin2 h
The actual solution of this equation exists
only if the right part of the last expression does
not exceed unity:
sin2 qc
1
sin2 h
ð3:72Þ
which corresponds to qc h. The solution is
1
sin qc
u ¼ arccos
;
2
sin h
1
sin qc
u ¼ p arccos
2
sin h
ð3:74Þ
B sin hðcos h þ tgqc sin hÞ þ Atgqc kc 0
ð3:75Þ
for
B
1
sin qc
u ¼ arccos
2
sin h
ð3:76Þ
sin h
þ Atgqc kc 0
cos qc
ð3:77Þ
Figures 3.11 and 3.12 depict the position of
the dangerous points on the well contour, where
the stresses reach their maximums. For small
inclination angles of the well, Fig. 3.11, they are
in the plane formed by the vertical and axis of the
well (points M).
As the inclination angle of the well grows, the
maximum shear stresses increase and as the angle
of inclination reaches the critical value h ¼ qC ,
a bifurcation occur: the maximums become
minimums and two additional pair of maximums
ð3:73Þ
Thus, for 0 h qc , extrema at points u ¼
0; u ¼ p correspond to maximums, and extrema
at points u ¼ p=2 correspond to minimums
(Fig. 3.11). For 0 qc \h p=2 additional
maximums
appear
at
the
points
h
i
sin qc
1
u ¼ 2 arccos sin h , and extrema at points u ¼
0; u ¼ p become minimums.
u¼0
qc h p=2
2
cos2 2u ¼
0 h qc
Fig. 3.12 Position of potentially dangerous points on the
well contour for large inclination angles
3.2 Mechanical Model of Stability of Inclined and Horizontal Wells …
49
– Starting from small values of the inclination
angle, with its increase the value of the
function Y grows in all points of the well
contour, reaching a maximum at point M
(corresponding u ¼ 0);
– at some angle of inclination, the value of the
function Y at point M becomes equal adhesion
sC , i.e. the shear stress in plane of weakining
at this point reaches the critical value;
– if the inclination angle of the well continues
to increase, the domain in which the Y reaches
the value sC increases. Therefore, as the
inclination of the well increases, the domain
in which the shear stresses reach the critical
value expands. When the size of this zone
increases so much that the rock reaches a state
of ultimate equilibrium, the wellbore wall
stability is lost;
– as the minimum inclination angle of the well
at which the loss of stability may begin, it is
naturally accept the angle corresponding to
reaching by the shear stresses at point M
strength ½s. According to (3.56) and (3.67)
this means that at this point the value Y becomes equal to the adhesion sC . The value of
the adhesion of 5 MPa corresponds to the
inclination angle of the well of about 50°;
– for a well inclination angle of more than 60°
the probability of failure is reduced. It’s
related to the fact that, as it is seen from
Fig. 3.12, that for angles greater than 60°, the
value of parameter Y near points M point
begins to decrease, resulting in a significant
reduction of the zone in which the shear
stresses reach the limit value.
Fig. 3.13 Dependence of combination of stresses Y on
polar angle u, for various well inclination angles h: green
line h ¼ 30 , blue line h ¼ 45 , orange line h ¼ 60 , red
line h ¼ 75
Thus, it can be concluded that the most dangerous from the point of view of well stability
loss are the inclination angles within 40°–60°
depending on the adhesion and the friction angle
of the rock. Let us emphasize once again that
reaching the critical value at one point on the
well contour by stress does not necessarily leads
to the beginning of wellbore failure. For the
failure to happen, the critical state must be
achieved within a sufficiently large domain.
Note that in addition to the fracture of the rock
along the weakening planes, in principle, another
mechanism is possible, associated with the
destruction of the rock under the influence of
maximum shear stresses on the planes that do not
coincide with the formation planes. Therefore, it is
necessary to consider both variants, and assume
occur, shifting form the points of the former
maximums in both directions along the circle
contour by the angle determined by expression
(3.76) (points M in Fig. 3.12) should be noted
that as the inclination of the well increases, not
only does the stress maximum increase, but also
the size of the high stressed domains increases.
This naturally increases the probability of failure.
Figure 3.13 depicts the dependence of distribution of the stress combination Y (3.67) on polar
angle u, for various angles of inclination of the
well h. For sandstones, the internal friction angle
qc is approximately 30°. However, along the
planes weakening (layering planes) the strength
properties of the rock are significantly reduced.
Therefore, calculations were made for internal
friction angle qc ¼ 15 . Calculations were made
for the well depth of 2900 m and drilling mud
density of 1.12 g/cm3. For the average density of
overlying rocks of 2.3 g/cm3 this corresponds to
rock pressure q = 66.5 MPa and bottom-hole
pressure pw ¼ 32:5 MPa.
Analysis of the results presented on Fig. 3.13
reviled:
50
3
Mechanical and Mathematical, and Experimental Modeling …
that the failure will occur according to the mechanisms for which the failure criterion be satisfied at
lower stresses (Chap. 1.2). However, it follows
from the above analysis that for the fracture
mechanism associated with the maximum shear
stresses, the stability of wells should not depend on
the angle of inclination of the well. It follows from
the fact that for satisfied relation (3.45), which is
the case of the majority of rocks, the stress distribution in the vicinity of an inclined borehole drilled
in transversal-isotropic medium coincides with
good accuracy with the solution for the borehole in
an elastic isotropic medium. Therefore, the value of
maximum shear stresses (or combination of stresses due to Druker-Prager or Coloumn-Mohr criteria) in the vicinity of the well does not depend on
the inclination angle of the well; so if the failure of
the well walls by this mechanism did not occur at
zero inclination angle of the well, it should not
occur at any inclination angle.
3.3
Stress State in the Vicinity
of the Well in Elastically
Anisotropic Rocks
While solving practical problems, it is important
to identify the nature and degree of anisotropy.
For weak anisotropy, the difference between the
results obtained according to models that account
and do not account for anisotropy becomes
insignificant. Quantitatively, the degree of anisotropy is determined in terms of some dimensionless parameters characterizing the properties
of the medium along various directions. In case
of permeability, thermal conductivity, electrical
conductivity and other properties characterized
by second-rank tensors, the components of which
are expressed through three independent values
(the principle values), as a value characterizing
the degree of anisotropy, it is natural to take the
ratio of maximum and minimum of the principle
values (or any function of this ratio). Thus, the
degree of anisotropy for permeability will be
characterized by the ratio of permeability in two
perpendicular directions along which it has
maximal and minimal values. In case of
transversal isotropy, which is inherent to layered
media such as sedimentary and metamorphic
rocks, the number of independent values determining the tensor properties of the second rank is
reduced to two. For the permeability tensor, these
are the permeability values in the isotropy plane
and the normal to it. The deviation of the permeability ratio along these directions from unity
will characterize the degree of anisotropy.
For the elasticity, characterized by a fourth-rank
tensor, the question of determining the degree of
anisotropy becomes less obvious. Even in the
considered case of transversal isotropy, the number
of independent constants determining elastic
properties is equal to five, and four independent
dimensionless combinations can be made of them.
Obviously, not all of them are suitable as measure of
the degree of anisotropy (e.g., Poisson’s ratios,
although dimensionless, do not determine the
degree of anisotropy). For the two-dimensional
problem (plane strain and plane stress), as shown in
(Lekhnitsky 1950, 1977), the degree (intensity) of
anisotropy is described by two independent
parameters. However, there is a desire to highlight
one, the most important parameter. Usually, for
transversally isotropic media, the ratio of modules
in the longitudinal and transverse directions is taken
as such a parameter (Batugin and Nirensburg 1972).
Although this choice appeared to be natural, it is
unlikely to be so unambiguous and not always
optimal, as will be demonstrated below.
Since the values of the greatest interest from
the practical point of view are stresses, the most
interesting among the values characterizing the
degree of anisotropy will be those characterizing
the deviations of the stress distribution compared
to the stresses distribution in isotropic medium
under the same condition, rather than formal
combinations of elastic constants. Since one can
hardly expect to find a universal combination
suitable for arbitrary geometries and boundary
conditions, it is logical to consider the most
characteristic, frequently encountered and as
simple problems. First of all, such problems
include the problem of stress concentration on
the contour of cylindrical wells (workings).
Below, the cases of wells in a transversally isotropic massif located within the plane of isotropy
and inclination to it will be considered.
3.3 Stress State in the Vicinity of the Well …
51
A Well with an Axis Lying Within the Isotropy Plane; Equi-component Initial Stress
State
The problem of a cylindrical hole with its axis
coinciding with one of the principle axes of the
elasticity tensor was solved by Lekhnitsky and
Soldatov (1961) [see also (Lekhnitsky 1950,
1977)]. Using this solution, the stress concentration on the well contour in a transversally
isotropic medium, the axis of which lies in the
isotropy plane and the medium is compressed by
1
hydrostatic stresses r1
xx ¼ rzz on infinity, can be
written as follows
f0 ð/Þ ¼
r//
E/
¼ 1þ
n k þ nðk 1Þ cos2 /
r1
E1
zz
o
þ ½ðk þ 1Þ2 n2 sin2 / cos2 /
ð3:37Þ
Here E/ is the modulus of elasticity in the
direction of tangent to the point of contour, E1
is modulus of elasticity in the direction of
x1 -axis.
1
E/ 4
¼ sin / þ m sin2 / cos2 / þ k2 cos4 /
E1
ð3:38Þ
Angle / is calculated from the x1 -axis
towards to x3 -axis.
Constants k; m; n are defined as
a33 E1
¼ ;
a11 E3
n2 ¼ 2k þ m
k2 ¼
m¼
2a13 þ a55
E1
¼
2m13 ;
a11
G13
ð3:39Þ
for plane stress, and
2b13 þ b55
b11
2a13 þ a55
a12 a13 2
¼
;
a11
a11
k2 ¼
b33
;
b11
12
Note that the second formula (3.40) ratios a
a11
13
and a
a11 make sense as Poisson’s ratios for planes
x1 x2 and x1 x3 , respectively. For rocks, the Poisson ratios are rarely greater than 0.3, and the
difference in their values in different planes is
hardly greater than 0.1. The value 2a12a11þ a44 is of
the order of unity (equal to two for the isotropic
body). Therefore, the contribution of the second
term in the second formula (3.40) is usually less
than one per cent comparing to the contribution
of the first term and in most cases is negligible.
According to solution (3.37), the extreme
stress concentrations on the contour are achieved
at the points N corresponding to the polar angles
/ ¼ 0; p and at the points M corresponding to
the polar angles / ¼ p=2 (Fig. 3.14), i.e. at the
principle axes of the elasticity tensor
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1
2k þ m 1
¼ 1þ
;
ð3:41Þ
k
kffi
pffiffiffiffiffiffiffiffiffiffiffiffiffi
fM ¼ 1 þ n k ¼ 1 þ 2k þ m k
fN ¼ 1 þ
fM fN for E1 E3 .
Typical distribution of stress concentrations
on the contour, corresponding to a pronounced
anisotropy, are shown in Fig. 3.15 for the
values aa1211 ¼ 0:2, aa1311 ¼ 0:2, a44 ¼ a55 ¼ kA
ða11 þ a33 2a13 Þ.
It follows from Eq. (3.37) that provided
nk1¼0
ð3:42Þ
σr
x3
σφ
M
K
m¼
φ
ð3:40Þ
x1
N
n2 ¼ 2k þ m
for plane strain conditions.
Here, constants b are compliance constants
modified for plane strain (1.18).
Fig. 3.14 Wellbore in anisotropic rocks and corresponding stresses
52
3
Mechanical and Mathematical, and Experimental Modeling …
Fig. 3.15 Distributions of stress concentrations on the well contour in transversally isotropic rock for the following
values aa1211 ¼ 0:2, aa1311 ¼ 0:2, a44 ¼ a55 ¼ kA ða11 þ a33 2a13 Þ; dashed line (long dashes) corresponds to k ¼ bb11 ¼ 1,
33
kA ¼ 0:5; dashed line (short dashes) corresponds to k ¼ bb11 ¼ 2, kA ¼ 0:5; solid line corresponds to k ¼ bb11 ¼ 2, kA ¼ 1;
33
33
dashed-dotted line corresponds to k ¼ bb11 ¼ 2, kA ¼ 2; dotted line corresponds to k ¼ bb11 ¼ 1, kA ¼ 2
33
33
the stress concentration on the contour is equal to
two, which coincides with the solution for isotropic media.
Condition (3.42) may be written down
through elastic constants as follows
a55 ¼
1
¼ a11 þ a33 2a13
G0
ð3:43Þ
for plane stress, and
1
¼ b11 þ b33 2b13
G00
ð3:44Þ
ða12 a13 Þ2
¼ a11 þ a33 2a13 a11
a55 ¼
for plane strain. Similarly (3.40) the last term in
(3.44) is usually negligible.
It follows from the above that a very significant parameter characterizing elastic anisotropy
is the deviation a55 ¼ G1
13 from the value
determined by (3.44) or (3.43). Thus, using the
ratio of the independent shear module G13 ¼ a1
55
to the shear module calculated by the formula
(3.43) appears natural
kA ¼
¼
G13
;
G0
G0 ¼ ða11 þ a33 2a13 Þ1
E 1 E3
E1 ð1 þ m31 Þ þ E3
ð3:45Þ
3.3 Stress State in the Vicinity of the Well …
53
The role of the independent shear module G13
on the value of the stress concentration on the
hole contour was noted by Lekhnitsky (1977).
More accurate, though less convenient, is the
parameter
kB ¼
G13
G00
ð3:46Þ
where G00 is defined by (3.44). The difference
between the parameters entered is usually
negligible.
For
kA \1,
which
corresponds
to
n k 1 [ 0, and the stress concentrations
have maximums in points of the intersections of
the contour with the principle axes of the elasticity tensor (points M, N Fig. 3.14), and minimums in points K, p4 \/\ p2 (exact values are
given by rather cumbersome expressions
obtained by equating derivatives of function
f 0 ð/Þ determined by the formula (3.37) to zero)
and symmetrical (with respect to the principle
axes of the elasticity tensor) points. For kA [ 1,
which corresponds to n k 1\0, the maxima
and minima change places. The approximation
for the maximum stress concentration for kA [ 1
may be obtained by developing the exact solution in a series over kA 1:
pffiffiffi
3 þ 2 k þ 3k 1 þ k2 2 bb13
11
fK ¼ 2 þ
pffiffiffi2
2
2 1 þ k ð1 þ k Þ
ð k A 1Þ
ð3:47Þ
The influence of parameter kA on the stress
concentration on the contour is illustrated by
Figs. 3.16 and 3.17. Figure 3.16 depicts the
dependences of stress concentrations fN , fM for
kA 1 on parameter kA corresponding to the
value of the parameters aa1211 ¼ 0:2, aa1311 ¼ 0:2,
a44 ¼ a55 ¼ kA ða11 þ a33 2a13 Þ: solid line
corresponds to fM ¼ fN for k ¼ 1; dashed line
corresponds to fM for k ¼ 2; dashed-dotted line
corresponds to fA for k ¼ 2. Figure 3.17 depicts
the dependence of stress concentrations at the
maximum points for the same parameter aij for
kA 1 (the lines for k ¼ 1 and k ¼ 2 and the
used parameters are indistinguishable, the
Fig. 3.16 Stress concentrations fN and fM for kA 1, a12 =a11 ¼ 0:2, a13 =a11 ¼ 0:2, a44 ¼ a55 ¼
kA ða11 þ a33 2a13 Þ: solid line corresponds to fM ¼ fN for k ¼ 1; dashed line corresponds to fM for k ¼ 2;
dashed-dotted line corresponds to fN for k ¼ 2
54
3
Mechanical and Mathematical, and Experimental Modeling …
Fig. 3.17 Stress
concentrations fM and fN for
kA 1
difference in numerical values was observed in
the fourth digits for kA ¼ 1:5).
An important fact, confirmed by the illustrations above, consists in deviation of the stress
concentration from that corresponding to the
isotropic case exists even for coinciding principle
values of the compressive modules ðk ¼ 1Þ. It is
seen in Figs. 3.15, 3.16 and 3.17 that for kA \1
the deviation of the stress concentration from that
corresponding to the isotropic case k from unity
leads to an increase in the stress concentration fN
and a decrease in the stress concentration fM
(however only under condition of kA 6¼ 1).
However, for kA [ 1 the ratio of modules k has
practically no influence (at least for the considered values of parameters) on the stress concentration, and leads only to a shift of position of the
maximum concentration towards to the direction
of the minimum modulus (Fig. 3.15).
Of course, the use of any pair of values
m; n; k directly as parameters characterizing
anisotropy is more rigorous mathematically, but
in addition to the parameter k, other parameters
are expressed through elastic constants by means
of rather cumbersome formulas and are deprived
of transparent meaning.
It is interesting to note that the combination of
elastic characteristics corresponding to the
fulfillment of the condition (3.43) [or, which is
the same, (3.45)] corresponds to one of the special cases considered by de Saint-Venant (1863)
[see also (Lekhnitsky 1950; Rabinovich 1946)],
for which the type of dependence of elastic
modulus on orientation in space has the most
simple form.
The condition (3.45) was described by Batugin
and Nierenburg (1972), as a condition for the
constancy of the directrix of the shear modulus in
the plane normal to the plane of isotropy of the
transversally isotropic materials. It is also shown
that this formula with an accuracy of 10% gives the
right values for 45 out of 47 considered rocks that
in the first approximation can be considered as
transversal-isotropic (siltstones, phyllites, shales,
sandstones, limestones, granites, granodiorites,
etc.). Values of elastic constants were taken from
experimental studies by different authors
(Lekhnitsky 1962; Skorikova 1965; Myachkin
1960; Rozovskiy and Zorin 1966; Sersembayev
1965; Clark 1942; Isaacson 1958; Belikov 1961).
In references to this paper, the formula (3.45) is
usually addressed as empirical, followed the
analysis of experimental data. Taking into account
the importance of this relation for determining the
stresses, as well as the studies of de Saint-Vienne,
this formula can hardly be considered as empirical.
3.3 Stress State in the Vicinity of the Well …
For 39 of the 47 rocks studied in Batugin and
Nierenburg (1972) (for other rocks, the set of
initial data on elastic constants was incomplete,
which did not allow us to carry out the required
analysis for them), the values of the ratio of the
independent shear modulus to the modulus calculated by the formulas (3.45), (3.44) were calculated—the relative difference did not exceed
2%. Parameters k; m; n for plane strain and values
of additives to the stress concentration calculated
by formulas (3.41), (3.47) are also provided. The
results show that for the majority of rocks, the
deviations of concentrations calculated according
to both (3.45) and (3.46) are negligible for
practical purposes. However, for one particular
rock the deviations in both cases are very significant. The values of deviation of stress concentration for the case of uniaxial compression
along the maximum module are also presented.
Note that condition (3.44) is used in the study
of wave propagation [Gassmann condition
(Gassmann 1964)] and corresponds to the ellipsoidality of the refraction surface. Fulfillment of
this condition with a good accuracy for the same
data set as in the present paper was investigated
in Annin (2009).
Hypotheses of an approximate fulfillment of
the conditions corresponding to remaining three
out of four particular subclasses of anisotropy
considered by de Saint Vincent (1863) were also
verified. These subclasses ðG4 ; F4 ; F2 Þ correspond to the ellipsoidal indicator surfaces in the
pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
spaces 4 ann ðnÞ; 4 Cnn ðnÞ; 2 Cnn ðnÞ (the case
considered above, G2 , corresponds to the ellippffiffiffiffiffiffiffiffiffiffiffiffiffi
soidal indicator surface in space 2 ann ðnÞ) (de
Saint-Venant 1863, 17; Pouya 2007) and appear
in models of damage mechanics and theories of
effective media; in addition, a number of analytical solutions have been obtained for these
subclasses (Pouya 2007). In the above expressions n is vector of normal in the corresponding
space; ann ðnÞ is compliance in the direction n;
Cnn ðnÞ is rigidity in the direction n; C ¼ a1 is
rigidity matrix, which is a reverse matrix of
compliance matrix. For these subclasses,
55
coefficients kA are also introduced, for which the
values a55 are defined as follows (Pouya 2007)
a55 ðG4 Þ
;
a55
pffiffiffiffiffiffiffiffiffiffiffiffi
a55 ðG4 Þ ¼ 2ð a11 a33 a13 Þ
k A ðG 4 Þ ¼
k A ð F4 Þ ¼
a55 ðF4 Þ
;
a55
2
a55 ðF4 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C11 C33 C13
ð3:49Þ
a55 ðF2 Þ
;
a55
C11 þ C33 2C13
a55 ðF2 Þ ¼
4
kA ðF2 Þ ¼
ð3:48Þ
ð3:50Þ
The above values for 39 rocks are presented in
Table 3.1.
It follows from the data analysis that the used
formula (3.45) gives the best accuracy, although
the differences in accuracies are not so great: the
P
mean deviations N1 Nn¼1 ðk 1Þ and standard
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
PN
2
1
, where N ¼ 39
ð
k
1
Þ
deviations
n¼1
N
and
k ¼ kA ; kB ; kA ðG4 Þ; kA ðF4 Þ; kA ðF2 Þ
are
equal to −0.031, −0.034, −0.036, −0.069,
−0.078, and 0.121, 0.121, 0.125, 0.135, 0.146,
respectively.
A slight difference in the accuracy of the
various formulas indicates that the anisotropy of
the rocks under consideration is not too pronounced, rather than a deep correlation between
the elastic constants given by various formulas.
The results of research (Batugin and Nirensburg 1972) and analysis of elastic solution
(Lekhnitsky and Soldatov 1961; Lekhnitsky
1950, 1977) for the stress concentration on the
circular hole (well) suggest that for the analysis
of stress state around the wells in anisotropic
rocks, in most cases we can use the Lamé’s
solution for isotropic body.
However, for those rare rocks for which
relation (3.45) is not satisfied with the necessary
accuracy, the use of the isotropic solution would
0.29
0.20
0.21
0.17
0.35
0.15
0.26
0.15
0.19
Fresh granite
Limestone
Marble
Sandstone
0.23
0.20
0.42
0.22
0.19
0.18
0.23
Rich sylvinite
Sandstone
Basalt 3
Gneiso-granite red
Basalt
0.18
0.24
0.19
0.47
0.18
0.17
0.16
Red-gray granite
0.20
0.16
0.35
Siltstone
0.15
0.29
0.21
Basalt 1
Zuber clean
0.18
0.41
0.35
Granite
0.10
0.08
Amphibolite
Poor sylvinite with zuber
0.29
−0.04
−0.03
−0.03
−0.03
−0.03
−0.02
−0.12
−0.03
−0.11
−0.03
−0.02
−0.04
−0.04
−0.05
−0.03
−0.03
−0.03
−0.03
−0.01
−0.02
−0.03
−0.02
−0.03
−0.05
−0.03
−0.06
−0.04
−0.08
−0.02
−0.08
−0.03
−0.10
−0.02
−0.09
−0.04
−0.05
−0.07
−0.06
−0.03
−0.02
−0.03
−0.19
−0.38
−0.01
a13
a12
0.56
0.44
0.48
0.52
1.15
0.44
0.40
0.45
0.39
0.48
0.38
0.69
0.39
0.93
0.63
1.05
0.25
0.69
0.69
0.28
0.48
8.33
a44
0.98
0.98
0.98
0.98
0.98
0.97
0.98
0.96
0.95
0.95
0.96
0.95
0.94
0.94
0.93
0.92
0.91
0.90
0.90
0.87
0.75
0.39
kA
0.98
0.98
0.98
0.98
0.98
0.97
0.97
0.96
0.95
0.95
0.95
0.94
0.94
0.94
0.93
0.92
0.91
0.90
0.89
0.86
0.75
0.38
kB
1.02
1.03
1.03
1.02
1.06
1.07
1.04
1.04
1.04
1.05
1.19
1.06
1.05
1.00
1.15
1.08
1.06
1.07
1.14
1.24
1.35
1.56
k
2.08
2.12
2.12
2.10
2.20
2.23
2.16
2.18
2.21
2.22
2.57
2.29
2.26
2.18
2.52
2.41
2.43
2.45
2.64
3.04
4.09
10.04
m
2.03
2.05
2.05
2.03
2.08
2.09
2.06
2.06
2.07
2.08
2.23
2.10
2.09
2.04
2.20
2.14
2.13
2.14
2.22
2.35
2.61
3.63
n
0.01
0.01
0.01
0.01
0.02
0.02
0.02
0.02
0.03
0.03
0.03
0.03
0.04
0.04
0.04
0.06
0.07
0.07
0.07
0.09
0.19
0.69
fN
0.01
0.01
0.01
0.01
0.02
0.02
0.02
0.02
0.03
0.03
0.03
0.04
0.04
0.04
0.05
0.06
0.07
0.07
0.08
0.11
0.25
1.07
fM
−0.02
−0.02
−0.03
−0.03
−0.03
−0.04
−0.03
−0.05
−0.06
−0.06
−0.06
−0.07
−0.07
−0.08
−0.09
−0.11
−0.13
−0.13
−0.14
−0.19
−0.37
−0.88
fK
−0.02
−0.03
−0.03
−0.02
−0.06
−0.07
−0.04
−0.04
−0.04
−0.05
−0.16
−0.06
−0.05
0.00
−0.13
−0.07
−0.06
−0.07
−0.12
−0.19
−0.26
−0.36
fN1
0.03
0.05
0.05
0.03
0.08
0.09
0.06
0.06
0.07
0.08
0.23
0.10
0.09
0.04
0.20
0.14
0.13
0.14
0.22
0.35
0.61
1.63
fM1
0.98
0.98
0.98
0.98
0.97
0.97
0.98
0.96
0.95
0.95
0.95
0.94
0.94
0.94
0.93
0.92
0.91
0.90
0.89
0.85
0.72
0.37
kA (G4)
0.97
0.96
0.95
0.97
0.92
0.93
0.96
0.95
0.94
0.94
0.93
0.93
0.92
0.90
0.91
0.86
0.84
0.88
0.87
0.82
0.69
0.35
0.96
0.95
0.97
0.92
0.92
0.96
0.95
0.94
0.94
0.92
0.92
0.92
0.90
0.89
0.86
0.83
0.87
0.86
0.79
0.64
0.31
kA (F2)
0.97
(continued)
kA (F4)
3
Sandstone
0.28
0.24
0.25
Limestone
Limestone
0.19
0.12
0.10
0.08
Peridotite
0.93
Sand slate
Slate chloride
1.92
a11
Solid
a33
Table 3.1 Elastic parameters of rocks
56
Mechanical and Mathematical, and Experimental Modeling …
0.32
0.14
0.13
Tufopeschanik
Filit 2
0.20
0.18
0.16
Hibinit
Brown granite
1.04
0.13
0.64
Plagiogranite
Sandstone
0.19
0.18
0.17
0.19
0.15
0.17
0.18
0.14
Magmatite
0.25
0.20
Filit 1
0.26
0.23
Granite
Siltstone
1.49
0.15
1.05
Gray granite
Sandstone
0.13
0.17
0.13
0.17
Granodiorite dark gray
0.16
0.12
0.20
a33
Granodiorite pink
0.12
0.15
Fine-grained granite
0.20
Coarse granite
Plagiogranite
a11
Solid
Table 3.1 (continued)
0.39
0.34
−0.05
−0.04
−0.04
−0.04
−0.04
−0.03
0.00
−0.03
0.32
0.34
0.44
1.92
−0.03
−0.18
−0.02
0.38
0.42
0.59
0.58
2.70
0.40
0.41
0.31
0.38
0.30
0.49
a44
−0.13
−0.04
−0.04
−0.03
−0.04
−0.01
−0.07
−0.05
−0.04
−0.03
−0.11
−0.03
−0.09
−0.03
−0.04
−0.03
−0.03
−0.03
−0.04
−0.03
−0.04
−0.04
−0.04
a13
a12
1.12
1.10
1.07
1.07
1.06
1.05
1.05
1.04
1.04
1.04
1.02
1.02
1.01
1.01
1.01
1.00
0.99
kA
1.12
1.08
1.07
1.07
1.06
1.05
1.05
1.04
1.04
1.03
1.02
1.02
1.01
1.01
1.01
1.00
0.99
kB
1.16
1.09
1.07
1.05
1.28
1.05
1.13
1.03
1.01
1.13
1.19
1.18
1.02
1.00
1.01
0.99
1.01
k
2.01
1.99
1.97
1.91
2.45
1.98
2.13
1.95
1.93
2.21
2.37
2.33
2.00
1.97
1.99
2.00
2.05
m
2.08
2.04
2.03
2.00
2.24
2.02
2.09
2.00
1.99
2.12
2.18
2.16
2.01
2.00
2.00
1.99
2.02
n
−0.07
−0.05
−0.04
−0.04
−0.03
−0.03
−0.03
−0.03
−0.02
−0.01
−0.01
−0.01
−0.01
−0.01
−0.01
0.00
0.01
fN
−0.08
−0.05
−0.04
−0.05
−0.04
−0.03
−0.03
−0.03
−0.02
−0.01
−0.01
−0.01
−0.01
−0.01
−0.01
0.00
0.01
fM
0.17
0.10
0.09
0.10
0.08
0.06
0.07
0.06
0.05
0.03
0.02
0.02
0.02
0.02
0.02
−0.01
−0.01
fK
−0.14
−0.08
−0.07
−0.04
−0.22
−0.05
−0.11
−0.03
−0.01
−0.11
−0.16
−0.15
−0.02
0.00
−0.01
0.01
−0.01
fN1
0.08
0.04
0.03
0.00
0.24
0.02
0.09
0.00
−0.01
0.12
0.18
0.16
0.01
0.00
0.00
−0.01
0.02
fM1
1.11
1.10
1.07
1.07
1.03
1.05
1.04
1.04
1.04
1.03
1.01
1.01
1.01
1.01
1.01
1.00
0.99
kA (G4)
1.06
1.04
1.03
1.02
1.00
1.02
0.97
1.01
0.98
1.02
1.00
0.99
0.98
0.97
0.96
0.93
0.96
kA (F4)
1.04
1.03
1.03
1.02
0.95
1.01
0.95
1.01
0.98
1.01
0.98
0.98
0.98
0.97
0.96
0.93
0.96
kA (F2)
3.3 Stress State in the Vicinity of the Well …
57
58
3
Mechanical and Mathematical, and Experimental Modeling …
result in underestimation of the stress concentrations. The case kA ¼ GG130 [ 1 is especially dangerous, because the maximum stress peaks occur
in the zones where the maximum shear stresses act
in the isotropy planes, which are usually planes of
weakening (Goodman 1980; Jaeger 1960; Jaeger
et al. 2007; Zobak 2007; Karev 2016).
Thus, on the one hand, for the majority of
rocks the parameter kA ¼ GG130 is approximately
equal to unity, which allows using the solution
for isotropic medium, on the other hand, for
rocks for which the deviation of the given
parameter from unity is essential, this parameter
becomes governing the stress state.
Extreme values of this expression always
correspond to points N, / ¼ 0; p and M, / ¼
p=2 for any values of elastic parameters
Non Equi-component Initial Stress State
It is generally accepted that one of the principle
stresses ðrzz Þ is align vertically and is determined
by the weight of the overlying rocks. The other
two are supposed to be independent
fN ¼ 1 þ
rxx ¼ kx rzz ;
ryy ¼ ky rzz
ð3:51Þ
where kx ; ky are the coefficients of lateral compression; they are often assumed to be equal:
kx ¼ ky ¼ k. In case of strong tectonics, one or
both coefficients may be greater than unity
(Jaeger et al. 2007).
The stress state near the well with its axis
coinciding with the direction of one of the principle stresses and lying in the isotropy plane is
described by Lekhnitsky’s solution (Lekhnitsky
and Soldatov 1961; Lekhnitsky 1950, 1977). The
stress concentration on the contour may be
recorded as
f ð/Þ ¼
r//
¼ f0 ð/Þ ð1 kÞf1 ð/Þ
r1
zz
ð3:52Þ
where f0 ð/Þ is determined (3.37), and f1 ð/Þ is a
concentration of stresses on the well contour due
to uniaxial compression along the horizontal axis
f1 ð/Þ ¼
E/
k cos2 / þ ð1 þ nÞ sin2 /
E1
ð3:53Þ
fN1 ¼ 1
k
fM1 ¼ 1 þ n
ð3:54Þ
For this case, the relative value of the independent shear modulus (3.45) ceases to be decisive for determining the stress concentrations and
their divergence from the isotropic case.
The stress concentration in the points N, / ¼
0; p and M, / ¼ p=2 is obtained by substitution (3.41), (3.54) in (3.52)
n1
1
nk
þ ð 1 kÞ ¼ 1 þ
;
k
k
k
n1
ð1 kÞð1 þ nÞ
ð3:55Þ
fM ¼ 1 þ
k
Formulas (3.55) show that, similar to the
isotropic case, the reduction of stress along x1
axis compared to the hydrostatic case leads to the
increase in the stress concentration at the points
N and the decrease in the stress concentration at
the points M.
Stress concentrations for 39 rocks are given in
Table 3.1. The data analysis suggests that not too
pronounced inequality of the initial principle
stresses the influence of anisotropy on stress
concentration for the majority of rock is
insignificant.
Inclined Wells
To find out the influence of parameter kA ¼ GG130
on stress distribution along the contour of a well
inclined to the principal axes of the elasticity
tensor (inclined well in transversally isotropic
rock with horizontally located plane of isotropy),
finite element calculations were carried out,
which demonstrated that under condition (3.45)
the deviation of the stress concentration from the
value for isotropic medium is less than 1%. The
discrepancy is supposed to be due to inherent
error of the used calculation procedure.
3.4 Physical Simulation of Conditions in the Vicinity …
3.4
Physical Simulation
of Conditions in the Vicinity
of Inclined and Horizontal Wells
in Anisotropic (Layered) Rocks
The developed mechanical and mathematical
model can be used to determine the most dangerous points on the contour of horizontal wells
drilled in formations with pronounced layering.
However, the approach to modeling the
stress-strain states occurring in the vicinity of a
horizontal well in a transversally isotropic reservoir differs significantly from that usually applied
to vertical wells. This is due to the fact that in the
case of a vertical well, all the points on its contour
are absolutely identical in terms of the stresses
acting in them for both isotropic and transversalisotropic reservoirs. That’s not the case for horizontal wells. In isotropic rocks, the stresses are
also constant along the well contour, in isotropic
rocks, they change significantly along the well
contour and depend on the elastic characteristics of
the rock, Fig. 3.15. As it was noted in Sect. 3.2, for
the values of parameter kA \1 defined by formula
(3.49) the stress concentrations have maximums at
the points of intersection of the contour with the
main axes of the elasticity tensor (M, N) and
minimums at the points p4 \/\ p2.
The circle depicted on Fig. 3.15 corresponds
to circumferential stresses along the contour of
the horizontal well in isotropic rocks. Radial
stresses rr are constant along the well contour
and are equal absolute value to the fluid pressure
in the well. Since the maximum shear stresses
acting on the well contour are equal to
ðrr rh Þ=2, they will be the highest in the
points M and N, respectively (Fig. 3.9).
The main difference in testing specimens
according to programs corresponding to points M
and N is that at point N the stress rh acts perpendicularly to the layering plane, and at point M
does parallel to it, Fig. 3.9. Therefore, rock
specimens have to be placed in the loading unit
of the TILTS accordingly.
With the decrease of fluid pressure in the well,
the radial stresses rr equal to this pressure will
also decrease at points M and N while the
59
circumferential stresses rh will increase as they
are proportional to the difference between the
rock pressure and the fluid pressure in the well.
A horizontal well is the ultimate case of an
inclined well and corresponds to the inclination
angle h ¼ 90 . Therefore, for a horizontal well,
drilled in permeable rocks, the ratios (3.68)–
(3.70), (3.76), (3.77) remain valid, where h ¼
90 is set.
For the shear stress acting in the plane of layering, for the shear stress according to (3.58) is
s ¼ ðq þ pw Þ sin 2u
ð3:78Þ
where angle u is calculated from the vertical axis
of the well cross-section.
Compressive stresses normal to the layering
plane, according to (3.59), are
sn ¼ ðq þ pw Þð1 cos 2uÞ
ð3:79Þ
According to (3.76), the shear stress on the
contour of the horizontal well ðh ¼ 90 Þ reaches
its maximum at the points shifted from the vertical axis of the well cross-section by angle
u¼
1 p
qC
2 2
ð3:80Þ
For the angle of internal friction qC ¼ 15 this
angle reaches 37:5 , for qC ¼ 30 it reaches 30 ,
for qC ¼ 45 it reaches 27:5 . Therefore, the
fracture on the well contour must first occur for
these angles. So to determine the critical values
of pressure drawdown, modeling should b conducted for the specimens cut out at different
angles relative to the bedding plane.
References
Annin BD (2009) Transversal-isotropic model of the
geomaterials. J Appl Ind Math (in Russian). 12(3):
5–14
Aoki T, Tan CP, Bamford WE (1994) Stability analysis of
inclined wellbores in saturated anisotropic shales. In:
Computer methods and advances in geomechanics:
proceedings of the eighth international conference on
computer methods and advances in geomechanics.
Morgantown, West Virginia, USA: 2025–2030
60
3
Mechanical and Mathematical, and Experimental Modeling …
Batugin SA, Nirengburg RK (1972) Approximate dependence between elastic rock constants and anisotropy
parameters (in Russian). Physico-technical problems
of mineral resources development, 1:7–11
Belikov BP (1961) Elastic and strength properties of
rocks. In: IHEM proceedings. A kisser, 43p
Blokhin BC, Terent’ev VD (1984) Wellbore stability
assessment method. Oil Ind 7:12–15
Clark SP (ed) (1942) Handbook of physical constants.
Geol Soc
de Saint-Venant (1863) Mémoire sur la distribution des
élasticités autour de chaque point d’un solide ou d’un
milieu de contexture quelconque, particulièrement
lorsqu’il est amorphe sans être isotrope (Deuxième
article). J Math Pures Appl Sér 2 8:257–430
Dinnik AN (1925) About the rock pressure and calculation of the round mine support. Engineer (in Russian)
Gassmann F (1964) Introduction to seismic travel time
methods in anisotropic media. Pure Appl Geophys
58:63–112
Goodman RE (1980) Introduction to rocks mechanics.
Wiley, New York
Isaacson E (1958) Rock pressure in mines. Mining
Publications Ltd., London, 212p
Isaev MI (1958) On stability of well walls during drilling.
Izvestiya Vuzoviya. Sir. (Sighs) “Oil and gas” 10
Jaeger JC (1960) Shear failure of anisotropic rocks. Geol
Mag 97:65–72
Jaeger JC, Cook NGW, Zimmerman RW (2007) Fundamentals of rock mechanics. Blackwell. MyiLibrary,
Malden, MA Oxford, 475p
Katsaurov IN (1972) Mountain pressure Vyspek. 2. Rock
mechanics. Nedra, Moscow
Karev VI, Klimov DM, Kovalenko YF, Ustinov KB
(2016) About the destruction of the sedimentary rocks
under the conditions of the complex three-axial stress
state (in Russian). Izv RAS MTT 5:15–21
Lekhnitsky SG (1950) Anisotropic body elasticity theory
of M-L: State Institute of Technology and Theory.
299p
Lekhnitsky SG (1962) Theoretical study of stresses in an
elastic anisotropic array near the underground ellipti-
cal section. In: Proceedings of VNIMI. Sat.,
45:110–118
Lekhnitsky SG (1977) Anisotropic body elasticity theory.
Science, Moscow, 415p
Lekhnitsky SG, Soldatov VV (1961) Influence of the
elliptical hole position on the stress concentration in
the elongated orthotropic plate (in Russian). Izv USSR
Acad Sci OTN Mech Mech Eng 1:3–8
Petukhov IM, Zapryagaev AP (1984) Stability of the
wells of different diameters depending on the rock
stress state (in Russian). Oil Ind 5:22–25
Pouya A (2007) Ellipsoidal anisotropies in linear elasticity extension of Saint Venant’s work to phenomenological modelling of materials. Int J Damage Mech.
16:95–126
Rabinovich AL (1946) About the elastic permanent and
strength of the aviation materials (in Russian).
Proc CAGI 582:1–56
Rozovskiy MI, Zorin AN (1966) Application of the
integrated operators to the determination of the
stresses and displacements of the underground structure contour taking into account the influence of the
time factor and anisotropy. In: Problems of rock
mechanics. Alma-Ata: science, pp 367–372 (in
Russian)
Rzhanitsyn BA, Tsarevich KA (1936) Chemical methods
of the oil well collapse control. Oil Ind 4 (in Russian)
Sersembayev AA, et al (1965) Rock mechanics research.
Alma-Ata Science
Skorikova MF (1965) On anisotropy of elastic properties
of rocks about. Sakhalin. Izv USSR Acad Sci Sir Geol
3
Spivak AI, Popov AN (1994) Rock destruction during
drilling of wells. Nedra, Moscow, 261p
Timoshenko (1937) Theory of elasticity (in Russian).
ONTI, Moscow, 508p
Timoshenko SP, Goodier J (1979) Theory of elasticity.
Science, Moscow, 560p
Vasiliev YN, Dubinina NI (2000) Stress state model of
the bottom-hole zone (in Russian). Oil Gas 4:44–47
Zobak MD (2007) Reservoir geomechanics. Cambridge
University Press, 443p
4
Equipment for Studying Deformation
and Strength Properties of Rocks
in Triaxial Loading
The development of hydrocarbon fields is a
complex problem, which requires knowledge and
experience accumulated in various fields of science and engineering practice. An integrated
multidisciplinary approach has become particularly relevant at the present stage characterized
on the one hand, the significant deterioration in
gas and oil reserves structure, and, on the other
hand, the creation of new well drilling and
completion technologies, a significant advancement in research and modeling of geomechanical
processes in the formation using the new
high-speed computers. To fill the models it is
necessary to know the properties of the objects of
study.
To determine the strength characteristics of
rocks, laboratory tests of rock specimens are
carried out on specialized devices. Methods for
studying deformation and strength soils properties are determined by State Standards. The main
groups of devices used to determine the deformation and strength soils characteristics are presented in Table 4.1.
One of the most common methods for determining deformation and strength characteristics
of rocks is the triaxial compression test, due to its
simplicity and efficiency.
Soil testing by triaxial compression method
according to State Standards GOST 12248-96,
ASTM D2850, ASTM D4767, BS 1377
(Table 4.2) is carried out to determine the following parameters of materials: strength and
deformability: the angle of internal friction,
© Springer Nature Switzerland AG 2020
V. Karev et al., Geomechanics of Oil and Gas Wells,
Advances in Oil and Gas Exploration & Production,
https://doi.org/10.1007/978-3-030-26608-0_4
cohesion, elastic modules and the Poisson ratio
for sands, clayey, organic, mineral, and organic
soils.
4.1
Karman Type Installations
Installations implementing the thriaxial compression method are based on the Karman principle, Fig. 4.1.
The test specimen has a cylinder shape. A load
is applied to the end faces along the axis of the
specimen, usually, by rigid plates and independently on the lateral surface of the cylinder, usually using a strong flexible casing filled with oil
under pressure (Hasbullah et al. 2018).
Thus, the Karman type installation is a conditionally triaxial loading unit, since despite the
loading is carried out on the entire surface of the
specimen, it is possible to control only two
components of force during loading: vertical and
radial.
Facilities based on Karman principle differ in
axial load, method and magnitudes of all-round
compression, size of tested specimens. These
devices allow:
• testing in automatic or semi-automatic mode;
• axial loading with rigid plates;
• applying all-round compression of the specimen by pressure of air or liquid;
• providing vertical load in steps or continuously at a given rate;
61
62
4 Equipment for Studying Deformation and Strength Properties …
Table 4.1 Devices for determination of deformation and strength soils characteristics
soils characteristics.
Compression device
Compression device with measurement
of lateral stresses
Device for compression testing of Device for pre compaction
soils in relaxation mode
Direct shear apparatus with kine- Direct shear apparatus with static shear
matic shear loading
loading
(continued)
4.1 Karman Type Installations
63
Table 4.1 (continued)
Tension testing device by spheri-
Triaxial compression device
cal indenter
True triaxial loading device
Table 4.2 Standards for triaxial testing
Test method
Russia
England
USA
Features
Unconsolidated undrained (LH)
GOST 12248-96, part 5.3
BS 1377, part 7
ASTM D2850
No pore pressure measurement
Consolidated Undrained (CN)
GOST 12248-96, part 5.3
BS 1377, part 8
ASTM D4767
With pore pressure measurement
Consolidated-drained (CD)
GOST 12248-96, part 5.3
BS 1377, part 8
–
With volume change measurement
64
4 Equipment for Studying Deformation and Strength Properties …
(a) with force control;
(b) with displacement control.
Both test methods can be used, but method
(a) is preferable because it allows to set and
maintain at a given level the dynamic affect
parameters (or change them according to a given
program) during the experiment.
Triaxial compression tests not only allow a
number of parameters to be determined for various soil models, but also allow the tests to be
carried out using various schemes and various
loading paths.
Fig. 4.1 Karman type installation
• providing fluid supply to the specimen from
below or above and its removal;
• providing pore pressure measurement at the
top and the bottom of the specimen;
• providing measurement of volumetric strains
of the specimen;
• providing measurement of radial and axial
strains of the specimen;
• providing filtration of the liquid through the
specimen.
The Karman facility should include: a
three-axis compression chamber; a device for
creating, maintaining and measuring pressure in
the chamber; a mechanism for vertical loading of
the specimen; devices for measuring vertical and
volumetric strains of the specimen; devices for
measuring pore pressure based on the compensation principle and pressure sensors of high rigidity; back pressure system. The design of the
three-axis compression chamber should provide:
lateral expansion of the specimen; water squeezing from the specimen; tightness of the main parts;
minimum possible friction of the stem in the
bushing of the chamber; measurement of the
volume of liquid pumped into the chamber.
There are two main options for dynamic triaxial tests:
4.2
True Triaxial Loading Systems
The most complete information on the properties
of rock is provided by using true triaxial loading
units (TTLU), which are not yet widespread. In
contrast to the conditionally triaxial loading
facilities, the TTLU allow to control the stresses
on three axes independently and simultaneously.
However, it is needed to note, this feature causes
certain difficulties at creating the installation.
There is no universal solution for building such
units so far, so there are no two identical TTLUs
in the world.
However, all TTLUs can be conditionally
divided into 3 types:
1. installations with hard plates;
2. installations with flexible plates;
3. mixed-type installations.
Every type
disadvantages.
has
its
advantages
and
1. Installations with rigid plates
The loading unit of this TTLU type consists of 3
hydraulic pistons, which transmit the load on the
faces of the specimen through rigid (metal)
plates. This type of installation allows to create
stresses much higher than the other two types of
installations, has the necessary stability of the
4.2 True Triaxial Loading Systems
loading system, provides to carry out different
loading paths on each of three axes and the
ability to test large specimens. This type of
installation can be divided into two subtypes:
(1) installations where the pressure plates move
only along the compression axis.
(2) installations where pressure plates can move
not only along the compression axis, but also
perpendicularly to it.
In the case of the first subtype, the pressure
plates have to be smaller than the specimen faces
to avoid touching of the adjustment plates while
loading, which creates edge effects. The second
subtype does not have this disadvantage due to
the possibility of the plates moving perpendicularly to the compression axis, but there is
another problem, friction between the specimen
and the plates, which leads to some measurement
error.
2. Installations with flexible plates
This type of installation in two or more directions
has flexible plates. Flexible plate has a shell
made of durable rubber that is filled with liquid
and takes the form of a specimen surface, thus
eliminating edge effects. However, these systems
must not be subjected to high stresses due to the
low strength of shell materials and it is also
impossible to achieve high stability of the loading system. There are restrictions on the size of
the specimens.
3. Mixed type installations
These units are a combination of the two previous types, and their capabilities are supplemented
by a study of the permeability of the rock specimen and temperature control of the specimen.
Of course, they cannot create such large stresses
as apparatuses of the type 1 or provide possibility
to avoid edge effects, as installations of the type
2, but they allow for a combination of both features at the certain level.
65
4.3
Examples of True Three-Axis
Loading Installations
Below are some examples of particular installations and the teams that address scientific issues
by using them (Kwasniewski et al. 2013).
1. Triaxial Trials Truly
University of Mons—FPMs, Mons, Belgium.
Heads are J.-P. Tshibangu and F. Descamps.
The installation is designed to study the effect of
complex loading on the behavior of rocks at great
depths (Descamps et al. 2012; Descamps and
Tshibangu 2008).
Installation is of type 1.
It develops pressure in each direction up to
500 MPa. Special specimens of 31 mm * 30 mm *
30 mm are made for the installation. The strength of
the machine is estimated at 3.2 MN/mm (Fig. 4.2).
Installation test method: this installation allows
for routine three-axis testing. The first step of test
is to increase all three stress values to the specified
level r1 ¼ r2 ¼ r3 . In the second step, stress r3
remains constant and the other two are increased
up to a specified level r1 ¼ r2 . In the third step,
only stress r1 increases up to the end of the test,
i.e. it increases to the limit state after which the
plastic deformation of the specimen begins. Then
the specimen is unloaded. This test can be performed with a variation of the Lode parameter
values from the conventional three-axis compression
ðr2 ¼ r3 Þ to three-axis expansion ðr1 ¼ r2 Þ.
2. Obayashi Corporation, Kiose, Tokyo, Japan
Head is K. Suzuki.
The unit is designed to study interaction between
cracked and undamaged rocks. The main features
of this installation are the large size of the
specimens and the ability to create large loads in
all three directions (Dexter et al. 2019). The size
of specimens varies from 500 mm * 500 mm *
500 mm to 700 mm * 700 mm * 700 mm.
66
4 Equipment for Studying Deformation and Strength Properties …
Fig. 4.2 Triaxial trials truly installation at the University of Mons, Belgium
Installation is of type 1.
The load created by the machine in various
directions:
• 5 MN on X, Y
• 10 MN on Z.
Installation test procedure.
Prior a test Teflon sheets (0.1 mm thick) with
silicone grease are attached to the rigid plates of
machine to reduce friction. Then a specimen is
isotopically squeezed in three axes until the
values r1 ; r2 ; r3 will not reach the required initial value. Various initial values may be selected:
0.1, 0.2, 0.4, 0.8 MPa. After reaching the desired
initial stresses, while keeping the horizontal
stresses constant, the specimen is sequentially
compressed in the vertical direction. Finally, the
specimen is unloaded.
3. Lassonde Institute and Department of Civil
engineering University of Toronto, Toronto,
Canada
Heads are R. P. Young, M. H. B. Nasseri and L.
Lombos.
The facility is used to study types of faults in
rocks induced by seismic activity, changes in
elastic properties, and fluid filtration (Goodfellow
et al. 2015; Bai et al. 2019). Filtration of liquid
through the specimen is carried out due to the
holes made in the rigid plates, through which the
liquid is supplied to the specimen faces by a
pump. This makes it possible to investigate permeability of the rock in three directions.
Installation is of type 3.
Load on the main axis is 6.8 MN, on the side
are 3.4 MN.
Specimen dimensions 80 mm * 80 mm *
80 mm.
16 piezoelectric inductors mounted on rigid
loading station plates in direct contact with the
surfaces of the specimen faces allow to study
high-frequency wave velocities and acoustic
emission. The plates are also equipped with a
temperature control system, which allows to heat
the specimen up to 200 °C.
4.3 Examples of True Three-Axis Loading Installations
67
Specimens for tests are made using a special
WasinoCNC grinding machine. The grinding
technology has been adapted so that the deviation
from the non-parallelism of the specimen faces is
no more 5 microns.
Testing procedure.
Experiments are performed using a unique
true triaxial geophysical imaging cell within a
custom made MTS polyaxial loading frame. First
a specimen is hydrostatically loaded at a speed of
0.0002 mm/s up to 5 and 10 MPa of effective
stress respectively. Ultrasonic wave velocity
measurements were simultanously carried out at
every 1–2 MPa of loading along all three axes
(one vertical nd two horizontal directions).
Acoustic emissions and a continuous wave form
streaming system were armed to events. At
5 MPa hydrostatic stress Flexible Rubber Membrane is activated, by applying 2 MPa seal
pressure to all 12 edges of the Cubic Skeleton
Rubber Seal enclosing the cubic specimen followed by directional permeability measurements
based on steady-state flow method. At 10 MPa of
hydrostatic stress keeping r1 at this value, r2 and
r3 are raised simultaneously under drained conditions to 20 MPa of stress. 3D permeability and
3D ultrasonic wave velocity are measured systematically. Next r1 is increased with the same
displacement rate along the main stress direction
(vertical axis) until failure and beyond. Acoustic
emission, wave velocity tomography, 3D
stress-strain and 3D directional permeability are
monitored according to the pre-designed testing
plan at various stress increments of r1 .
Specimen sizes range from 50 mm * 50
mm * 50 mm to 200 mm * 200 mm * 200 mm.
The unit is also equipped with acoustic sensors,
which allows to record acoustic impulses occurring due to the formation and growth of cracks.
Installation is of type 3.
The hydraulic fracturing modeling procedure is
as follows. A 5 cm rock cubic specimen is prepared
for hydraulic fracturing experiment using the
TTSC. A hole is drilled in the center of the specimen and the fluid is injected into an open section of
approximately 1.5 cm in the middle of the cube.
Honey is used as the injecting fluid for this test and
is applied at a constant flow rate of 100 cl/h. The
external vertical and horizontal stresses are applied
to the specimen. In the regime when the vertical
stress essentially exceeds the horizontal stresses,
the fracture is expected to propagate in a vertical
plane. Also, large horizontal stress anisotropy is
considered to ease the propagation of the fracture
along the maximum stress direction. Besides notches are made inside the wellbore to help initiation
of the induced fracture.
4. Department of Petroleum Engineering, Curtin
University, Perth, Australia
Head is V. Rasouli.
This unit allows to study models where hydraulic
fracturing and sand production take place during
well operation. The working volume of the installation was made so that it was possible to conduct
experiments in a wide range of specimen sizes
(Rasouli et al. 2013; Gholami and Rasouli 2013).
4.4
Triaxial Independent Loading
Test System TILTS
TILTS is a unique test system of triaxial unequal
component loading, created at the Institute of
Mechanics Problems of the Russian Academy of
Sciences (Fig. 4.3) and designed to study deformation, strength and filtration characteristics of
rocks of oil and gas, ore and coal fields. Specimens for test are cubes with edge 40 or 50 mm.
The system is an electro-hydraulic testing
machine with an automated control system
(ACS). The facility allows the load to be controlled both by force and displacement. This
makes it possible to conduct a test up to complete
destruction of the specimen. The permeability of
the specimen is measured automatically
throughout the test. The forces are measured by
strain gauges, the displacements are measured by
inductive sensors, and the permeability is determined by the flow meters.
68
4 Equipment for Studying Deformation and Strength Properties …
Fig. 4.3 Triaxial
independent loading test
system (TILTS)
The ability of TITLS to load the specimen
independently on each of the three axes makes it
possible to reconstruct during the experiments
any stress states occurring in the bottom-hole
formation zone during well drilling, completion,
and operation, and to study influence of stress
state on the filtration properties of the rock
(Karev and Kovalenko 2013).
Such ability is available due to the original
kinematic scheme used in the design of the
loading unit, which allows the pressure plates to
supply the stresses to the whole faces of specimens without creating obstacles to each other
(Figs. 4.4, 4.5 and 4.6).
Permeability is determined on the flow rate of
air passing through the specimen and supplied a
compressor. For this purpose, one pair of pressure plates has channels for the supply of compressed gas to the specimen and for the exit f gas
filtered through the specimen and perforation for
the uniform supply and exit of gas over the
specimen face (Karev et al. 2016).
The specimen is being prepared for measuring
permeability along one of the axes in conditions of
complex stress state as follows. The specimen axis
along which the gas will be filtered is selected as
required. Four faces of the specimen parallel to the
filtration axis are being covered with a latex shell
Fig. 4.4 Pressure plates in initial position
Fig. 4.5 Pressure plates after specimen deformation
4.4 Triaxial Independent Loading Test System TILTS
69
Fig. 4.6 Loading unit cross-section after specimen
deformation
or aqueous polyvinyl acetate solution polymerization at room temperature, Fig. 4.7.
The latex shell is being dried at room temperature for several hours. The shell created on
the side faces in this way is being made thin
enough, not more than 50 µm, so as not to
introduce a significant error in the results of
measuring specimen strains. And at the same
time, such a shell has sufficient strength and
elasticity to ensure the tightness up to the formation of macro cracks in the specimen.
The Automatic Permeability Measurement
System (APMS) is used directly for permeability
measurement during the experiment, which
allows continuous monitoring of the permeability
change during the specimen testing on TILTS.
APMS is equipped with two flow meters that
allow to measure an air flow rate in a wide range:
from 0.5 ml/min to 5 l/min, and also two digital
pressure gauges measuring pressure at the inlet
and the outlet of the specimen. Signals from the
flow meters and pressure gauges are transmitted
to the controller of the automated control system,
processed, displayed on the monitor and recorded to computer memory.
Preliminarily the range of pressure values p at
the inlet of the specimen is set, for which the
dependence of air flow rate Q on the difference of
squares of inlet and outlet pressure Q ¼
Q p2 p2a is linear (pressure at the outlet or the
specimen is always atmospheric pa ). The air
Fig. 4.7 Specimen covered with latex shell
pressure during the test of the specimen is set in
this interval, where the gas flow is described by
Darcy law. Then the permeability coefficient is
determined by
k¼
2lQl
Fpa ðp02 1Þ
ð4:1Þ
where l is dynamic air viscosity, l is length of
the specimen, F is cross-sectional area of the
specimen, p0 ¼ ppa .
In general, l depends on the content of water
vapor, industrial oil vapor in gas and temperature. As the control of water vapor and oil vapor
concentration in gas is associated with significant
material costs and technical difficulties, APMS is
equipped with filters—dehumidifiers that remove
water and oil vapor from the gas. To determine l
at the controlled temperature, tabular data were
used. Taking into account the fact that TILTS is
located in a laboratory room with a sufficiently
stable temperature throughout the whole process
of testing one specimen, and the gas supply to
TILTS is carried out through copper pipes of
relatively long length, and the volume flow rate
of gas does not exceed 20 l/min, it can be
accepted with a sufficient degree of accuracy that
the gas temperature is constant and equal to the
air temperature in the room.
70
4 Equipment for Studying Deformation and Strength Properties …
References
Bai Q, Tibbo M, Nasseri HB, Paul Young R (2019) True
triaxial experimental investigation of rock response
around the mine—by tunnel under an in situ 3d stress
path. Rock Mech and Rock Eng. https://doi.org/10.
1007/s00603-019-01824-6.
Descamps F, Tshibangu J-P (2008) Development of an
automated triaxial system for thermo-hydro-mechanical
testing of rocks J.-P. ARMA 08:197–210
Descamps F, da Silva M, Schroeder C, Verbrugge J
(2012) Limiting envelopes of a dry porous limestone
under true triaxial stress states. Int J Rock Mech Min
Sci 56:88–99
Dexter P, Henke KR, Simon AC, Yarbrough LD (2019)
Rock Mechanics. In book: Earth Materials, 493–513
Goodfellow S, Nasseri MHB, Lombos L, Paul Young R
(2015) A triaxial hydraulic fracture experiment ISRM
13th International Congress on Rock Mechanics.
Montreal, Quebec, Canada
Gholami R, Rasouli V (2013) Mechanical and elastic
properties of transversely isotropic slate. J Rock Mech
Rock Eng 47(5):1763–1773
Hasbullah N, Dayu A, Riska E, Khairurrijal K (2018)
Axial and lateral small strain measurement of soils in
compression test using local deformation transducer.
J Eng Tech Sci 50(1):53–68
Karev VI, Kovalenko YuF (2013) Triaxial loading system
as a tool for solving geotechnical problems of oil and
gas production. In: True triaxial testing of rocks. CRC
Press, Balkema, Leiden: 301–310
Karev VI, Klimov DM, Kovalenko YuF, Ustinov KB
(2016) Anisotropic rock destruction model under a
complex loading. Phys mesomech 19(6):34–40
Kwasniewski M, Xiaochun Li, Takahashi L (2013) CRC
Press, 384p
Rasouli V, Pervukhina V, Müller TM, Pevzner R (2013)
In-situ stresses in the Southern Perth Basin, the
Harvey -1 well site. Exploration Geophys 44(4):289–
298. https://doi.org/10.1071/EG13046
5
Loading Programs for Rock
Specimens on Triaxial Independent
Loading Test System (TILTS)
5.1
Determining Strength
and Elastic Characteristics
of Rocks
The program of triaxial testing of rock specimens
is aimed at determining the stresses required to
fracture the rock specimen at various levels of
comprehensive compression (Klimov et al.
2010). To determine the required parameters, a
fracture stress value must be determined for at
least three comprehensive compression pressures. The difficulty lies in the fact that due to the
shortage of core material taken from great depths
and significant heterogeneity of rocks, it is
desirable to conduct all experiments on a single
specimen.
TILTS allows all three tests to be carried out
with different values of all-round compression on
the same specimen within the same experience.
In other words, each experience consists of three
loading cycles of the specimen. From cycle to
cycle, the pressure of the specimen’s pre-uniform
crimping increases, followed by an increase in
one stress at the same two other stresses.
During the loading process, the deformation
of the specimen is continuously monitored and
the current Young tangent module is calculated
in the direction of the increasing stress. The
specimen is loaded until the Jung tangent module
has decreased in the direction of the increasing
stress compared to the maximum recorded in this
test cycle by about 70–75%. Then the specimen
is unloaded to the current all-round compression
© Springer Nature Switzerland AG 2020
V. Karev et al., Geomechanics of Oil and Gas Wells,
Advances in Oil and Gas Exploration & Production,
https://doi.org/10.1007/978-3-030-26608-0_5
on the first and second cycles, and on the third
cycle the specimen is loaded until the specimen
is destroyed.
Note that the specimen load should be controlled by movement, not load. Otherwise, it is
almost impossible to stop the loading of the
specimen in the plastic region in time, without
bringing it to failure. The TILTS three-axis
independent loading test system makes it possible to perform such works.
The implemented test program of the specimens allowed to obtain three limiting states at
various stresses of all-round compression and
calculate the strength characteristics of the rock
—adhesion module and internal friction angle.
Below are the test programs for each load
cycle.
Cycle One
The loading program during cycle 1 is shown in
Fig. 5.1. Cycle 1 consists in three stages.
At the first stage, the specimen is compressed
along all three axes up to stresses of 2 MPa.
At stage 2, the stresses in two directions
remain constant and equal to 2 MPa, and in the
third direction (usually axis 2 of the loading unit)
coinciding with the vertical axis of the core, the
stress continues to grow at a constant rate of
strains, usually 4 10−6 s−1. The strains in three
axes are measured and diagram r–e along the
loading axis is determined. For each point of this
curve, the current tangent modulus Dr/De is
determined by means of linear approximation of
data of 21 experimental points (10 points before
71
72
5
Loading Programs for Rock Specimens on Triaxial Independent …
Fig. 5.1 Loading program. Cycle 1
Fig. 5.2 Loading program. Cycle 2
and 10 points after the point in question).
Loading at stage 2 terminates when the tangent
modulus fells down to the value of 25–30% of
the tangent modulus at the initial, linear section
of the curve r–e.
After that, the specimen is unloaded along the
2-axis down to the value of 2 MPa, so that at the
end of the first cycle the specimen is brought to
the state of uniform all-round compression by
stresses of 2 MPa.
Cycle Two
The specimen test program during Cycle 2 is
shown in Fig. 5.2.
Loading in Cycle 2 is similar to loading in
Cycle 1, with the only difference that in the
second step all three stresses applied to the
specimen in three axes rise evenly from 2 MPa
up to 10 MPa.
Then the specimen is loaded along one direction (axes 2) with the constant strain rate (the
same as in cycle 1). When the value of the tangent
module decreases down to the value of 25–30% of
the tangent modulus at the initial, linear section of
the curve r–e, the specimen is unloaded along the
2-axis down to the value of 10 MPa.
Fig. 5.3 Loading program. Cycle 3
Cycle Three
The specimen test program during Cycle 3 is
shown in Fig. 5.3.
5.1 Determining Strength and Elastic Characteristics of Rocks
Loading in Cycle 3 is similar to loading in
cycle 1 and cycle 2, with the difference that in the
third step all three stresses applied to the specimen in three axes rise evenly from 10 MPa up to
20 MPa. Then two stresses are kept equal to
20 MPa, and the stress along the axis 2 increase
at a constant strain rate. But unlike the first and
second cycles, loading during Cycle 3 does not
stop, but continues until the specimen failure.
After obtaining the deformation curves,
Young’s moduli and Poisson’s ratios are determined within the range of elastic deformation,
and parameters of plasticity model, adhesion s0
and internal friction angle q0 , are determined by
means of the construction of the Mohr’s circles.
The methodology for rock strength characterization based on triaxial testing is shown
below, using the example of a specimen from
Kirinskoye gas condensate field. Cubic specimen
with a rib of 40 mm was made of core
(medium-grained sandstone) taken at a depth of
2776.
Figures 5.4, 5.5 and 5.6 show the specimen
deformation curves for each of the three cycles
(Kovalenko et al. 2011).
73
Fig. 5.5 Deformation curves of specimen K1 during the
second load cycle
Fig. 5.6 Deformation curves of specimen K1 during the
third cycle of loading
Fig. 5.4 Deformation curves of specimen K1 during the
first cycle of loading
On the basis of the experimental results of the
specimen the Mohr circles were constructed for
each of the test cycles (Fig. 5.7), then the
strength characteristics of the investigated sandstone were obtained: adhesion s0 = 6.8 MPa,
internal friction angle q0 = 36.1°.
74
5
Loading Programs for Rock Specimens on Triaxial Independent …
Fig. 5.7 Mohr’s circles, based on the tests. Specimen tests K1
5.2
Programs for Physical Modeling
of Deformation Processes
in the Vicinity of Inclined
and Horizontal Wells
in Isotropic and Anisotropic
(Layered) Formations
Uncased Borehole in Isotropic Formation
Under Hydrostatic Rock Pressure
Permeable Rock
Values of stresses sr ; s/ ; sz in the vicinity of the
well in a permeable formation are given by the
ratios (3.12) and (3.13).
The loading program corresponding to the drop
of pressure pw at the bottom-hole is shown in
Fig. 5.8 (Klimov et al. 2008, 2009, 2013). Here
stresses S1, S2, S3 refer to the axes of the TILTS
loading unit, with the increasing stress S2 being the
loading parameter. Stress S2 corresponds to the
circumferential stress s/ , stress S3 corresponds to
Fig. 5.8 The “well” program for permeable rocks
radial stress Sr, and stress S1 is equal to the vertical
rock pressure SZ at a given depth. Loading is carried out in three stages.
Stage 1: The specimen is hydrostatically
compressed along three axes to the stress equal to
the difference between rock pressure q at the
depth h and fluid formation pressure p0 (OA
5.2 Programs for Physical Modeling of Deformation Processes …
section in Fig. 5.8). Point A corresponds to the
stresses acting in the rock skeleton before drilling
the well.
Stage 2: At the ABi sections, one stress component (S2) continues to grow, the second (S1)
remains constant, and the third (S3) decreases, with the
load varying so that the average stress s ¼ ðs1 þ s2 þ
s3 Þ=3 throughout Step 2 remains constant.
The end point of the stage (point B) corresponds to the state when the well is drilled and
filled with technical water.
On sections ABi according for (3.13)
s2 ¼ 2ðq þ p0 Þ, s1 ¼ q þ p0 , s3 ¼ 0.
Stage 3: The third stage simulates the process
of pressure decrease in the well (sections of BC
in Fig. 5.8). As can be seen from formulae (3.13),
the circumferential and vertical stress are
increasing, but the latter is increasing approximately twice as slowly.
At the third step the loading continues until
the specimen breaks down or the stresses reach
the values corresponding to the maximum possible drawdown (full borehole dehydration).
In the course of the experiment, the strains of
the specimen in three directions and the permeability of the specimen along one of the axes are
measured.
Impermeable Rock
In impermeable layers, the stress acting in the
skeleton are equal to the total stress acting in the
formation, i.e. si ¼ ri and are given by formulae
(3.14) and (3.15). The loading program corresponding to stresses (3.15) is presented in
Fig. 5.9. Here, as in the program for permeable
Fig. 5.9 Well program for impermeable rocks
75
rock, stress s1 ; s2 ; s3 refer to the axes of the
TILTS loading unit and correspond to total stress
rz ; r/ ; rr . Specimens were loaded in two
stages.
Stage 1: The specimen is compressed uniformly on all faces up to stresses equal to rock
pressure q at a given depth. Point A corresponds
to the stresses acting in the rock before drilling
the well. At point A s1 ¼ s2 ¼ s3 ¼ q.
Stage 2: The second stage of loading (sections
of ABi) simulates the stress states occurring in
the vicinity of the well for different values of
bottom-hole pressure at a given depth, i.e. at
different values of mud density. At sections ABi,
one of the components (S2) continues to grow,
the second (S1) remains constant, and the third
(S3) decreases; the load changes in such a way
that the average stress s ¼ ðs1 þ s2 þ s3 Þ=3
throughout Step 2 remains constant. For sections
ABi according to (3.15) s2 ¼ 2q þ pw , s1 ¼ q,
s3 ¼ pw .
Each point on section AB3 corresponds to a
certain downhole pressure; i.e. for a given depth
corresponds to a certain value of mud density.
In the course of the experiment, the strains of
the specimen is measured in three directions.
Perforation Hole in Isotropic Massif Under
Hydrostatic Rock Pressure
Expressions for stress in the vicinity of the perforation hole are given in Sect. 3.1, where it was
noted that the stress state near the walls of a
perforation hole at distances small comparing to
its length can be accurately approximated by the
formulae (3.12) and (3.13) for an uncased well.
Thus for modeling deformation processes in this
zone during change of pressure in a well it is
possible to use the loading program shown on
Fig. 5.10 (Klimov et al. 2003).
As for the stresses occurring in the vicinity of
the tip of the perforation hole, they can be well
approximated by the stresses acting in the
vicinity of the spherical cavity. Their expressions
are given by the relations (3.20) and (3.21).
The loading program corresponding to the
decrease of pressure pw on the bottom-hole for
the tip of the perforation hole is shown in
Fig. 5.10. Here, stress s1 ; s2 ; s3 refer to the axes
76
5
Loading Programs for Rock Specimens on Triaxial Independent …
Fig. 5.10 Loading program for modeling stress state
near a perforation hole
of the TILTS loading unit and correspond to the
effective stress su ; sh ; sr .
The loading program consists of three stages.
Stage 1: The specimen is compressed uniformly on all faces up to stresses equal to the
difference between rock pressure q at depth h and
the reservoir fluid pressure p0 (point A at
Fig. 5.10). Point A corresponds to the stresses
acting in the ground skeleton before drilling the
well.
Stage 2: Two stress components (s1 and s2 )
corresponding to the circumferential stress
increase equally and the third stress s3 corresponding to the radial stress decreases (sections
AB in Fig. 5.10). The average stress s ¼
ðs1 þ s2 þ s3 Þ=3 remains constant. The end of
the second stage corresponds to the stress state
around the perforation holes before well completion. At point B, stresses are given by (3.21).
Stage 3: The two components of the stresses
continue to grow, and the third component
remains practically zero. Stage 3 models the
change in stress state near the perforation hole
when the pressure in the well drops.
In the course of the experiment, three components of strains and the permeability along one
axis are measured.
Uncased Borehole in Isotropic Formation
Under Uneven Natural Rock Pressure
Below is an example of a loading program for a
particular specimen and the results of the test.
A direct physical modeling of the process of
rock deformation and fracture in the vicinity of a
horizontal well drilled through the payout zone
of the Vostochno-Messoyakhskoye field of PJSC
Gazprom Neft was performed using TILTS unit
when a drawdown was created in it. The well
was drilled along the direction of the maximum
horizontal initial stress at density of
qm = 1.4 g/cm3.
A cubic specimen 40 40 40 mm was
made of core specimen from depth of
h = 848.3 m. The following values of the absolute values of vertical and horizontal initial
stresses in the reservoir obtained on the base
of geophysical studies and calculations were
used:
Vertical stress jqV j = 16.9 MPa;
= 15.3 MPa;
Maximum horizontal stress qmax
H
min
Minimum horizontal stress qH = 13 MPa;
Formation fluid pressure p0 = 7.9 MPa.
The loading program was built for point N in
Fig. 3.10. It’s presented on the Fig. 5.11. Stresses s1 ; s2 ; s3 refer the axes of the loading unit of
TILTS installation, with the increasing stress s2
being the loading parameter. Stress s2 corresponds to the circumferential stress s/ , stress s3
corresponds to radial stress sr , and stress s1
corresponds to stress sz for a given depth.
The loading program consists of four stages.
Stage 1: At the first stage, the specimen is
compressed evenly in three axes up to the values
corresponding to the initial stresses in the reservoir:
s1 ¼ qmax
s 2 ¼ j qV j p0
p0 = 7.4 MPa;
H
= 9.0 MPa; s3 ¼ qmin
=
5.1
MPa (sec
p
0
H
tions OA in Fig. 5.11). Points A correspond to
the stresses acting in the rock skeleton before
drilling the well.
Stage 2: The second stage of loading (AB
sections) simulates the process of over balance
5.2 Programs for Physical Modeling of Deformation Processes …
Fig. 5.11 Loading program; modeling borehole in the
reservoir under uneven initial rock pressure
drilling of the well. Each point on sections AB
corresponds to a certain bottom-hole pressure
greater than the formation fluid pressure, i.e. a
certain amount of the over balance. One of the
stress components (S2) continues to grow and the
other two decrease. The terminate point of the
stage (point B) corresponds to the state when the
well is drilled and the bottom-hole pressure
corresponding to the drilling mud density
qm = 1.4 g/cm3 is equal to pw ¼ qm h = 11.9
MPa. At point B from (3.32) we have
s1 ¼ 3:4 MPa; s2 ¼ 13:9 MPa; s3 ¼ 0.
Stage 3: The third stage simulates the process
of downhole pressure decrease down to the value
of reservoir fluid pressure (sections BC in
Fig. 5.11). At point C pw ¼ p0 and according to
(3.32) s1 ¼ 7:4 MPa; s2 ¼ 21:9 MPa; s3 ¼ 0.
Stage 4: The fourth stage simulates the process of drawdown, i.e. further decrease of
bottom-hole pressure until drying out (section
CD in Fig. 5.11). As the drawdown grows, the
stress s3 remains practically equal to zero, the
stress s1 and s2 increase.
In the course of the experiment the strains of
the specimen was measured in three directions,
permeability was measured in the horizontal
plane of the specimen, i.e. along axis, which
corresponds to the radial direction of the well.
77
Fig. 5.12 Deformation curves; modeling borehole in the
reservoir under uneven initial rock pressure
Fig. 5.13 Change in permeability during loading
The initial permeability of the specimen was
42.5 mD.
The deformation curves of the specimen in
three axes (e1 ; e2 ; e3 —corresponding components of strains) are given on Fig. 5.12, the
change in the permeability of the specimen k is
given on Fig. 5.13.
The figures demonstrate that the specimen
was deformed almost elastically and did not
fractures even under the stresses corresponding
to the complete drainage of the well ðpw ¼ 0Þ,
point D in Fig. 5.12. The permeability of the
specimen has dropped to almost zero.
78
5
Loading Programs for Rock Specimens on Triaxial Independent …
Loading Programs for Modeling Inclined
and Horizontal Wells in Anisotropic
(Layered) Media
As noted above, the most dangerous from the
point of view of initiation of fracture is the
vicinity of point M, Fig. 3.15. Therefore, for
modeling the conditions in the vicinity of the
well inclined at various angles, the loading program corresponding to the change of the principal stresses at point M with changing pressure in
the well was chosen. For the elastic constants of
the rocks under consideration the axes of the
principal stress at point M coincide with the axes
of the well (Chap. 3). In other words, the specimen edges should be subjected to the effective
stresses corresponding to stresses Sr ; S/ ; Sz on
the well contour at point M. Moreover, according
to FEM simulation the stress distribution around
inclined wells within the accuracy of calculations
also does not differ from the distribution of
stresses in isotropic media (Chap. 3). Therefore,
hereafter, for stresses in the vicinity of an
inclined well drilled in a transversally isotropic
formation, the solutions for the well in an elastic
isotropic medium (p.3.1) will be used.
Inclined Well
As the inclined wells are drilled mainly in the
host rocks, which are impermeable, effective
stresses are equal to total stresses, i.e. according
to (3.8) si ¼ ri . The corresponding test program
is shown in Fig. 5.14.
Stresses si refer to the axes of the TILTS
loading unit; the stress usually increases along
axis 2, i.e. the stress s2 is the so-called loading
Fig. 5.14 Loading program for inclined borehole
parameter. Stress s2 corresponds to circumferential stress r/ , Fig. 3.3. The stress s3 corresponds to radial stress rr , i.e. equal to the fluid
pressure in the well, the stress s1 corresponds to
stress rz , i.e. the rock pressure at a given depth.
The loading program consists of two stages.
Stage 1: The specimen is compressed uniformly up to stresses equal to rock pressure q at
depth h. Point A corresponds to the stresses
acting in rocks before drilling the well.
At point A s1 ¼ s2 ¼ s3 ¼ jqj ¼ ch, where c
is the average density of overlying rocks, h is the
depth.
Stage 2: The second stage (sections ABi)
simulates the stress states arising in the vicinity
of the well at different values of bottom-hole
pressure at a given depth, i.e. at different values
of mud density. Each point on ABi section corresponds to a certain value of bottom-hole pressure, i.e. to a certain value of mud density for a
given depth. For the states corresponding to mud
densities of interest, the specimen is kept for a
long time at a constant load to register creep
deformation.
At sections ABi, one stress component (S2)
continues to grow, the second (S1) remains constant, and the third (S3) decreases, the loads
changing so that the average stress s ¼
ðs1 þ s2 þ s3 Þ=3 remains constant. The values of
the stresses applied to the faces of the specimen
are s1 ¼ jqj, s2 ¼ 2jqj pw , s3 ¼ pw . Loading
continues until the specimen failure.
The strains of the specimen are measured in
three directions throughout the experiment.
Horizontal Well
As mentioned above (Chap. 3), for the majority
of rocks the stress distribution around horizontal
wells in anisotropic rocks is practically the same
as the stress distribution in isotropic environments. Therefore, the values of stresses applied
to the faces of the specimen in the course of
experiments are determined by formulae (3.13).
The corresponding test program is presented
in Fig. 5.15.
Stage 1: The specimen is compressed hydrostatically over all faces up to stresses equal to the
difference between the rock pressure q and the
5.2 Programs for Physical Modeling of Deformation Processes …
79
Fig. 5.15 Loading program for horizontal well
Fig. 5.16 The scheme of “hollow cylinder” experiment
formation fluid pressure p0 (section OA). Point A
corresponds to the stresses acting in the ground
skeleton before drilling the well. At point A
s1 ¼ s2 ¼ s3 ¼ jqj p0 .
Stage 2: During the second stage of loading
(sections of ABi), one stress component (S2)
continues to grow, the second (S1) remains constant, and the third (S3) decreases, so that the
average stress s ¼ ðs1 þ s2 þ s3 Þ=3 throughout
step 2 remains constant. The terminate point of
the stage (points Bi) corresponds to the
bottom-hole pressure equal to the formation fluid
pressure:
S2 ¼ 2ðjqj p0 Þ; s1 ¼ jqj p0 ; s3 ¼ 0
Stage 3: The third stage simulates the process
of further reduction of pressure in the well (sections BiCi). Stress S3 (corresponding to radial
stress sr in the rock skeleton remains almost zero,
stresses S2 and S1 corresponding to the stresses
s/ and sz increases, but S1 increases two times
slower. Loading continues until the specimen
failure. Three principle strains and permeability
along one direction are measured.
5.3
Hollow Cylinder
The standard scheme of the “hollow cylinder”
experiment in particular, is presented on
Fig. 5.16 (Klimov et al. 2010). Cylindrical
specimens are produced with a length of about
12 cm and a diameter of about 5 cm, With holes
of 8–10 mm diameter drilled in the middle of
each cylinder. The specimens are then placed
into a rubber jackets and loaded.
Loading is carried out according to the Karman scheme, i.e. the specimen is loaded by
uniform lateral compression and axial compression. The air is blown through the specimen,
which carries the particles, whose weight is
measured during the test.
During the test, the lateral compressive load
on the specimen is increased at a constant rate
until sand is detected. The moment of the
beginning of sand production is fixed and then
the weight of the produced sand is continuously
measured.
TILTS installation also allows for specialized
experiments on cubic specimens with a central
hole, which can be considered as analogous to
the known hollow cylinder tests carried out on
Karman-type machines.
However hollow cylinder testing on TILTS
has significant advantages over conventional
testing:
– TILTS allows the cubic specimen to be loaded independently on each of the three axes
in any loading program, including modeling
the actual stresses acting in the formation;
– During tests, measures of the specimen strains
in three directions allows independent
80
5
Loading Programs for Rock Specimens on Triaxial Independent …
Fig. 5.17 Assembling of the loading unit of TILTS installation for “hollow cylinder” tests
detecting the beginning of the failure of the
hole walls by deviation stress-strain dependencies from linearity.
For these tests specially manufactured specimens with a center holes and pair of loading
plates with center channels were used, Fig. 5.17.
The tests are carried out on cubic specimens with
a rib length of 40 or 50 mm, in which center
holes of 10 mm diameter are drilled. In the
course of the experiment, air at a pressure of
about 0.1 MPa is supplied through the channel in
the upper active loading plate, which coincides
with the hole in the specimen.
The air passing through the channel in the
specimen leaves through the channel in the lower
loading plate, which coincides with the opening
in the specimen. Through this channel and a tube
attached the sand transmits to an electronic scale
connected to a computer. The accuracy of the
electronic scales is 0.001 g and the recording is
done every 2 s.
The specimen loading was stepwise. At each
step, the load was increased by 10 atm on each
axis, followed by a 5-min period of constant
load.
The beginning of sand production corresponds to the beginning of rock destruction on
the hole walls.
The “hollow cylinder” experiments clearly
demonstrate the influence of the type of natural
stress state on the character of deformation and
fracture in the vicinity of the well with lowering
the pressure on the bottom-hole.
Below are the results of such modeling for a
horizontal well for two cases:
– the well is drilled in a reservoir that is under
uniform all-round compression by the rock
pressure;
– the well is drilled in a formation that is under
uneven compression by rock pressure: the
vertical stress is higher than the stresses in
horizontal plane. In such cases the ratio of
horizontal and vertical stresses is referred to
as lateral support coefficient. In the experiment, the lateral support coefficient was
assumed to be 0.4.
The results of test according to program
“hollow cylinder” on a specimen under uniform
hydrostatic rock pressure are presented on
Figs. 5.18 and 5.19.
The results of the test of the specimen with
lateral support of 0.4 are shown on Figs. 5.20,
5.21 and 5.22. Figure 5.20 depicts the specimen
loading program, composed on the basis of the
relations of n. 3.1 for the hydrostatic rock
pressure.
We emphasize that the tested specimen was
made of the same piece of core as the previous
one.
Figure 5.21 depicts the dependence of the
mass sand production from the hole on the value
5.3 Hollow Cylinder
81
Fig. 5.18 Sand production, hydrostatic compression
Fig. 5.19 Deformation curves of the specimen, hydrostatic compression
of vertical compression of the specimen. The
deformation curves of the specimen are shown
on Fig. 5.22.
Photos of the tested specimens tasted with
lateral support of 1.0 and 0.4, respectively, are
Fig. 5.20 Loading program, lateral support 0.4
given on Figs. 5.23 and 5.24. It is seen from the
pictures that under uneven rock pressure rock
destruction occurs in the vicinity of the hole, and
hence, the situation is drastically different from
that occurring under hydrostatic rock pressure.
82
5
Loading Programs for Rock Specimens on Triaxial Independent …
Fig. 5.21 Dependence of mass sand production on the vertical compression, lateral support 0.4
Fig. 5.24 Wellbore destruction, lateral support 0.4
References
Fig. 5.22 Deformation curves, lateral support 0.4
Fig. 5.23 Wellbore destruction, hydrostatic loading
Klimov DM, Kovalenko YuF, Karev VI (2003) Implementation of the method of georgeline to increase
injectivity of injection wells (in Russian). Tehnologii
TEK 4:59–64
Klimov DM, Kovalenko YuF, Karev VI, Usachev EA
(2008) On the need to take into account the strength
characteristics of rocks in determining the optimal
spatial position of the well (in Russian). Drill and
petroleum 10:18–21
Klimov DM, Karev VI, Kovalenko YuF, Ustinov KB
(2009) On the stability of inclined and horizontal oil
and gas wells (in Russian). Actual problems of
mechanics. Mechanics of deformable solid. Ishlinsky
Institute for problems in Mechanics RAS, Nauka,
Moscow, 455–469
References
Klimov DM, Ter-Sarkisov RM, Chigay SE, Kovalenko YF,
Ryzhov AE (2010) Determination of strength characteristics of rocks Shtokman GKM and assessment of risks
of sand removal during its development (in Russian).
GAS Industry of Russia 11:57–60
Klimov DM, Karev VI, Kovalenko YuF, Ustinov KB
(2013) Mechanical-mathematical and experimental
83
modeling of well stability in anisotropic media. Mech
Solids 48(4):357–363
Kovalenko YuF, Kharlamov KN, Usachev EA (2011)
Borehole stability of the middle Ob region (in
Russian). Tyumen-Shadrinsk, 174p
6
Dependence of Permeability
on Stress State
The role of stresses occurring in oil and gas
formations is not limited by their influence on
well stability during drilling and processing.
Influence of stress state on permeability of rocks
forming reservoir and, as a consequence, on
productivity of wells is also of great importance.
Despite understanding of the importance of
this issue by the scientific society, the issue has
been devoted a negligible attention so far both in
mathematical modeling and developing simulators of field processing.
One of the main reasons for this is that the
permeability of rocks depends not only on the
values of stresses in them, but also on the type of
stress state.
When considering the influence of stress state
on permeability it is usually supposed that the
influence is reduced, at least at first approximation, to a dependence of permeability on the
volumetric stress, i.e. on the first invariant of
stress tensor. However experiments performed in
the Geomechanics Laboratory of IPMech RAS
(Karev and Kovalenko 2013a, b; Klimov et al.
2015; Karev et al. 2016) suggest that the main
contribution to the change in permeability is due
to shear stresses rather than due to hydrostatic
compression. The underlying physical mechanism is assumed to be the following. The permeability is determined by a system of connected
channels, mainly in the form of cracks with rough
faces. The even all-round compression results in
normal stresses on the crack faces, the latter due
to their roughness may not close significantly.
© Springer Nature Switzerland AG 2020
V. Karev et al., Geomechanics of Oil and Gas Wells,
Advances in Oil and Gas Exploration & Production,
https://doi.org/10.1007/978-3-030-26608-0_6
Contrary, the externally applied shear stress
results in appearing relative displacements of the
crack faces. That displacements in case of weak,
plastic rocks lead to crumping of the faces,
compaction and closure of cracks, and as a result,
to decrease in permeability. In hard brittle rocks,
relative displacement of crack faces may lead to
dilatancy and increase in permeability. In both
cases, further growth of applied stresses (of any
type but hydrostatic compression) may lead to
growth of new cracks accompanied by increase in
permeability (see Chap. 10). Therefore, for
modeling, as the first approximation, the dependence of permeability on stress state was chosen
in the form of the dependence of permeability on
the intensity of shear stresses, i.e. the second
invariant of the deviator of stress tensor (multiplied by a scalar constant).
It is impossible to calculate this dependence, it
can be established only experimentally and then
include into appropriate models. But in order to
carry out such research, experimental facilities
have to create real three-dimensional stress states
in rock specimens and determine permeability
during a test.
A large cycle of studies of the influence of the
stress-strain state on the permeability for different
types of rocks from reservoirs of oil and gas
fields was carried out on the TILTS according to
the method described above. These works have
allowed to establish that permeability of rocks
essentially depends on stresses. Depending on
the type of rock and the values of stresses, the
85
86
permeability can both decrease and increase, and
these changes can be irreversible. It should be
noted that in the development of enhanced oil
recovery methods this factor has not been taken
into account so far, although under certain conditions it can be decisive for the selection of the
optimal parameters for drilling, completion and
operation of wells.
Oil and gas reservoirs in most cases are
formed by carbonate (limestone, dolomite) or
terrigenous rocks (sandstone, siltstone, argillite)
with varying clay content. A research was carried
out on the properties of rocks from reservoirs of
oil and gas fields of various lithological composition, lying at depths from one hundred meters
to four kilometers, with various coefficients of
formation pressure anomalies.
Most of the research was carried out on fields of
Western Siberia, Kama, Volga, and Kuban regions:
Symoriakhskoye, Shushminskoye, Lovinskoye,
Vat-Yeganskoye, Tevlino-Russkinskoye (“LukoilWestern Siberia”), Siberian (“Lukoil-Perm”), Kislorskoye, Kurraganskoye, Vostochno-Perevalny,
Ikilorskoye, Cheremukhovskoye, Yenorusskinskoye
(RITEK),
Novo-Pokurskoye,
Yuzhno-Lokosovskoye (Slavneft), Ulyanovskoye,
Kaminskoye, Sykhtynglorskoye, VostochnoSurgutskoye
(Surgutneftegaz),
SeveroDolginskoye, Karmalinskoye (Gazprom). The
depths of these fields are 2000–3000 m, their
reservoirs are mainly terrigenous rocks with various, often quite high, clay content.
Rocks from large fields with carbonate reservoirs
at depths of about four kilometers were also tested:
Tengiz, Astrakhan gas condensate field (AGCF),
Urengoy gas condensate field (UGCF). These fields
are characterized by abnormally high reservoir
pressures (with abnormality coefficient up to 2).
Summarizing the results of tests on core
material taken from oil and gas reservoirs using
by TILTS, the rocks can be divided into three
categories according to the influence of stress
state on permeability.
Classification of Rocks According to the
Influence of Stress on Permeability
The first category is formed by rocks of densely
cemented fine-grained sandstones, argillites,
6
Dependence of Permeability on Stress State
dolomites, etc. Deformation of these rocks is
purely elastic within wide range of applied
stresses. Their permeability decreases with the
growth of stresses, however remains reversible,
i.e., after the stress removing, permeability
returns to its initial value. The properties of these
rocks are illustrated by Figs. 6.1, 6.2, 6.3, and
6.4. Figures 6.1 and 6.2 present the results of
testing the specimen from the reservoir of
North-Dolginskaya field (Barents Sea shelf);
Figs. 6.3 and 6.4 present the results of testing the
specimen from Achimov fields of the Urengoy
gas condensate field (UGCF). Figures 6.1 and
6.3 depict the programs of loading corresponding
to an open wellbore (Chap. 5.2, Fig. 5.8) and
dependences of permeability on time.
Fig. 6.1 Loading
program
North-Dolginskaya field
and
permeability;
Fig. 6.2 Deformation curves; North Dolginskaya field
6
Dependence of Permeability on Stress State
Fig. 6.3 Loading program and permeability; UGCF
Fig. 6.4 Deformation curves; UGCF
During the experiment, the deformation of the
specimen in three directions and permeability in
the layering plane were measured. Figures 6.2
and 6.4 depict deformation curves of the specimens during the test. Ordinate-axis corresponds
to parameter of loading—the principle stress r2 ,
corresponding to circumferential stress.
The specimen from North-Dolginskaya field
was collected from depth of 3017 m, which
corresponds to the rock pressure of about
70 MPa, the oil reservoir pressure being 30 MPa.
Figure 6.2 demonstrates that the specimen
was deformed almost elastically throughout the
experiment, which indicated by very small
residual strains after unloading. The permeability
of the specimen gradually decreased slightly
while loading (Fig. 6.1) both at the stage of
all-round compression and at the stage of shear
loading, which is obviously caused by the
87
decrease in the cross-section of filtration channels. The maximum reduction in permeability
was 30%, but it was almost completely recovered
on unloading.
A similar situation was observed when testing
a rock specimen from Achimov fields of the
Urengoy gas condensate field. The depth of
sampling was 3825 m, which corresponded to
rock pressure of 88 MPa, the reservoir fluid
pressure was abnormally high: 60 MPa.
It is seen from the deformation curves presented in Fig. 6.4 that the specimen deformed
elastically, there are practically no residual
deformations, the permeability during loading
has fallen slightly (by 30%), and it recovered
when unloading. In this case, with increase in
stress r2 above 100 MPa, which corresponds to
shear stress s = 50 MPa, small gradual increase
in permeability was observed, which is appeared
to be related to dilatancy: when the faces of
existing micro cracks that form the system of
filtration channels move relative to each other,
their opening may occur due to roughness.
The second category is formed by fine and
medium grained sandstones with a low content of
clay, siltstones and limestones. These rocks also
deform elastically under changes of stress states,
corresponding to minor and moderate pressure
drawdowns, with permeability unchanged or
slightly reduced. When the pressure drawdown
reaches a certain value, which depends on rock
properties, conditions of occurrence of reservoir,
pore pressure and other factors, the rock begins
to deform in elastically under a constant load (to
creep). As the inelastic deformations grow, the
permeability of the rock decreases significantly
(by tens and even hundreds percent). This drop in
permeability is irreversible, i.e. permeability
remains low when the stress is relieved.
With further increase in shear stresses (with
increasing pressure drawdown) the creep rate
increases, and when the deformation reaches
some critical value, the rock begins to fracture,
which is accompanied by a sharp increase in its
permeability even compared to the original
value. The nature of the fracture may vary. In
stronger rocks, specimens are usually destroyed
by several macro-cracks. Less resistant rocks,
88
6
Dependence of Permeability on Stress State
such as low strength sandstones, are disintegrated
into grains (sand).
Figures 6.5, 6.6, 6.7, 6.8, 6.9 and 6.10 present
results of experiments on specimens from terrigenous reservoirs of Symoriakhskoye (Western
Fig. 6.8 Deformation curves; Cheremukhovskoye field
Fig. 6.5 Loading program and permeability; Symoriakhskoye field
Fig. 6.9 Loading program and permeability; Ikilorskoye
field
Fig. 6.6 Deformation curves; Symoriakhskoye field
Fig. 6.7 Loading program and permeability; Cheremukhovskoye field
Fig. 6.10 Specimen deformation curves; Ikilorskoye
field
6
Dependence of Permeability on Stress State
Siberia), Cheremukhovskoye field (Tatarstan),
Ikilorskoye field (Western Siberia) that illustrate
properties of rocks of this category.
Tests were carried out under the program
simulating an open wellbore. The loading programs and dependency of permeability on time
are shown on Figs. 6.5, 6.7 and 6.9; deformation
curves are shown on Fig. 6.6, 6.8 and 6.10.
According to the lithological description,
reservoir rock of Symoriakhskoye field is formed
by coarse-grained clay-containing sandstones.
Initial permeability of the specimen was k0 = 9.4
mD. The specimen had been extracted from
depth of 2223 m; rock pressure at this depth was
51 MPa, fluid pressure in the field was close to
hydrostatic pressure, 21.5 MPa.
At the first stage of loading the specimen was
hydrostatically compressed up to 29.5 MPa. At
the second stage stress r2 reached value of
77 MPa, which corresponds to pressure drawdown of 9 MPa; then the specimen was completely unloaded. During the first stage, the
permeability of the rock has been reduced by
about 30%, which is obviously due to compaction. Further analysis of strains and permeability measured during the test showed that at
the value of loading parameter r2 = 70 MPa
(corresponding to pressure drawdown of about
6 MPa) the specimen began to creep, and the
permeability gradually fell by about half of its
initial value.
This can be explained by the infiltration of the
channels due to the presence of clay. At
r2 = 77 MPa, the deformation has completely
passed to the plastic stage (the deformation took
place under constant load), which was accompanied by a sharp increase in permeability. The
specimen was destroyed after the test, the latex
shell on the side edges was damaged, so the
permeability could not be determined after
unloading. Residual deformations were at the
level of 0.5%.
Results of testing of a rock specimen of
medium-grained
sandstone
from
Cheremukhovskoye oil field are presented on Figs. 6.7
and 6.8. The specimen possessed high initial
permeability k0 = 780 mD. The depth of sampling was 982 m; the rock pressure was
89
22.6 MPa; the fluid pressure was 9.8 MPa; the
value of initial effective stresses was approximately 12.8 MPa.
The specimen was deformed elastically until
the loading parameter r2 reached value of
35 MPa, then it began to creep, and at r2 ¼ 44
MPa, strains start to grow under the constant
load. The specimen was then unloaded in reverse
order of the loading program. Residual deformations were approximately 0.5%.
At the stage of evenly all-round compression,
the permeability fell slightly, by about 30%, and
then, with an increase in shear stresses, it continued to decrease, especially with the onset of
inelastic deformation: it dropped down to the
value three times lower than the initial value.
After transition to plastic deformation there
was a sharp increase in permeability up to 200%
of the initial value. It fell to 150% when unloaded. The applied loads resulted in appearance of
a system of micro-cracks in the specimen, which
resulted in an irreversible increase in its
permeability.
Similar behavior was manifested by a much
less permeable (k0 = 3 mD) rock specimen of
Ikilorskoye field (Figs. 6.9 and 6.10). The depth
of sampling was 2254 m; the rock pressure was
51.8 MPa; the reservoir pressure was 22.5 MPa;
the value of initial effective stresses was
approximately 29.3 MPa.
On reaching by the loading parameter the
value of 72 MPa, corresponding to 6.7 MPa
pressure drawdown, the specimen began to
deform plastically. After unloading the visual
observation reviled two macro cracks. Residual
deformations were about 0.3%.
The permeability of the specimen prior to
transition to plastic deformation has decreased by
about 50%, and as a result of plastic deformation
it has grown by more than 250% compared to the
initial value. After unloading, the permeability
remains 170% of the initial one.
The third category is formed by sandstones
and siltstones with high clay content. These
rocks already begin to “creep” intensively with
strong decrease in permeability under stress
changes corresponding to minor pressure drawdowns. However, even with significant
90
6
deformations, specimens do not collapse, they
continue to deform at a constant rate (like plastilin), and permeability permanently decreases.
Figures 6.11 and 6.12 depict the results of
rock of Nizhnechutinskoye (Komi) field. This
field is a very shallow (only 100 m) formation,
composed mostly by clay, with a small initial
permeability, 3.5 mD. Rock pressure at this
depth is 2.3 MPa; formation fluid pressure is
1 MPa; initial effective stresses are 1.3 MPa.
In the course of the experiments, the permeability fell by 60% at the stage of evenly
all-round compression (supposed due to narrowing of the filtration channels) and then by
another 50% under the increase in shear stresses,
which is supposed to be related to flooding of the
filtration channels. After unloading, the permeability has not recovered.
Experiments on another clay specimen from
the reservoir of the Karmalinskoye field (Kuban)
are presented in Figs. 6.13 and 6.14. The specimen was taken from interval of 2502–2508 m;
the rock pressure at this depth was 57.6 MPa; the
fluid formation pressure was 23 MPa; the value
of the initial effective stresses was 34.6 MPa.
The initial permeability of the specimen was
132 mD, during the evenly all-round compression it has decreased by about 80%. As shear
stresses increased, the permeability fell by
another 5–15% of the initial value and then
increased up to 30%, which might be explained
by dilatancy. When the value of the loading
parameter exceeded 100 MPa, the plastic deformation of the specimen began and its
Fig. 6.11 Loading
program
Nizhne-chutinskoye field
and
Dependence of Permeability on Stress State
Fig. 6.12 Deformation curves; Nizhnechutinskoye field
Fig. 6.13 Loading program and permeability; Karmalinskoye field
permeability;
Fig. 6.14 Deformation curves; Karmalinskoye field
6
Dependence of Permeability on Stress State
permeability gradually fell down to zero. The
specimen did not failed or destroyed as a result of
the test, but was severely deformed (by 2%).
Choosing the Optimal Technological
Parameters of Well Treatment on the Basis
of Rock Properties Studying
The results of the tests of various rock specimens
by using TILTS allow us to draw a number of
practically important conclusions.
For wells drilled in rocks of the first category,
the influence of stresses on filtration characteristics of the reservoir is not great and might not
be taken into account when selecting the modes
of operation at the well.
However, that is not the case for rocks of the
second and third categories. The ability of rocks
of the second category to deform intensely (“to
creep”) with a decrease in permeability under
shear stresses occurring in the bottom-hole zone
of the well can lead to a significant drop in the
flow rate of the wells.
When the pressure drawdown increases around
the open hole or perforation holes, a zone of
reduced permeability, a kind of low-permeability
“plug”, is formed. It is important to note here that
the process of rock deformation and permeability
decrease develops in time. Therefore, the reduction
of the well flow rate also occurs gradually in time.
With further increase in pressure drawdown,
when the deformation reaches some critical value,
the rock may start to fracture. As a result, an
artificial branched crack system, playing the role
of a new filtration channel system appears in the
well vicinity. This leads to a sharp irreversible
increase in permeability of the bottom-hole zone,
and filtration properties of rock may not only
restored, but also be significantly improved.
The described phenomenon of rock fracture
and irreversible increase in its permeability by
means of creation of necessary stresses in the
borehole zone became the basis of a new method
of increasing productivity of oil and gas wells—
the method of directional unloading of formation
—developed in the Institute for Problems in
Mechanics of the Russian Academy of Sciences.
It will be described in detail below in Chap. 10.
91
The effect of irreversible increase in permeability was observed during tests of specimens of
rocks from many fields, in particular, from a
reservoir of Symoryakhskoye field of
“LUKOIL-West Siberia”. Simulation of pressure
drawdown growth in open borehole using TILTS
(see above) has shown that under the stress state
corresponding to pressure drawdown of 5–
6 MPa the rock starts to deform inelastically,
which was accompanied by a noticeable decrease
in permeability. When the load reaches values
corresponding to pressure drawdown of 9 MPa,
the rock is fractures, disintegrated and the permeability increases essentially. The results
obtained during the completion of well
No. 7197 of this field confirmed the dependence
found by the laboratory experiments on specimens. In the process of well completion, pressure
drawdowns of 3, 6, and 9 MPa were created
sequentially, and after each steps the well productivity was determined according to methodic
of the level recovery curve. At pressure drawdown of 6 MPa, the productivity of the well has
dropped by about one and a half times as much
as the productivity determined at pressure
drawdown of 3 MPa. When the pressure drawdown reached 9 MPa, productivity increased
significantly compared to the initial one and after
finishing the well development remained four
times higher than had been expected.
The process of rock fracturing can be intensified by creation of perforation slots, holes, etc.,
resulting in significant increase in stresses acting
in the bottom-hole zone. Moreover, it is possible
to initiate the process of crack formation in
strong, tough rocks.
One example is the terrigenous reservoir of
the Siberian field in Perm region, CJSC
LUKOIL-Perm. Siberian oil-bearing formation is
formed by strong fine-grained sandstones.
Experiments on specimens using TILTS
demonstrated that the modeling of large pressure
drawdowns in open boreholes did not lead to
fracture and noticeable change in permeability
(see Figs. 6.15 and 6.16). However, during the
simulation of perforation holes in the open
borehole, which was achieved by drilling holes
of 8 mm diameter in the specimens, the rock
92
began to creep and fracture under stresses corresponding to high pressure drawdowns
(Figs. 6.15 and 6.16).
This can be explained by the lack of level of
stresses occurring in the vicinity of the open
borehole even at high pressure drawdowns. Perforation holes playing the role of stress concentrators significantly increase the stress acting in
their vicinity and initiate the process of crack
formation. The established dependencies were
confirmed during pilot field tests of the method
of directional unloading of the formation on the
wells of the Siberian field. Workover of the well
with the creation and long-term maintenance of
pressure drawdown close to maximum, did not
result in an increase in permeability. Similar
Fig. 6.15 Loading program and permeability; Siberian
field
Fig. 6.16 Deformation curves; Siberian field
6
Dependence of Permeability on Stress State
workover of injection well No. 310, but with a
preliminary perforation of the open hole, allowed
increasing the injectivity of the well from 8 up to
200 m3/day.
The test results presented in Figs. 6.17 and
6.18, illustrate another effect that was identified
in the course of research: fatigue failure. Three
loading cycles were performed on the specimen
during testing. In the first cycle the specimen
started creeping at the value of the maximum
principle stress corresponding to the circumferential stress in the well vicinity 96 MPa. The
specimen was then slightly unloaded in the
reverse order to the loading program and loaded
again. At repeated loading the creep started
Fig. 6.17 Loading program and permeability; Siberian
field
Fig. 6.18 Deformation curves of a specimen with a hole;
Siberian field
6
Dependence of Permeability on Stress State
already at 86 MPa. At the third cycle of loading,
the creep started at even lower stress—82 MPa.
Thus, under cyclic loading the critical stress
corresponding to elastic-inelastic transition
decreases with each succeeding cycle.
This explains the efficiency of methods of
enhancing oil recovery used in industry that are
based on creation of cyclic pressure drawdown at
bottom-holes.
Carbonate rocks of deep reservoirs with
abnormally high reservoir pressure such as the
Tengiz field in the Caspian Region should be paid
a special attention. At depths of 3.5–4 km, the
rock pressure reaches 90–100 MPa, but due to
abnormally high fluid pressure reaching 60–
80 MPa, the effective initial stresses acting on the
reservoir rock skeleton are relatively low, and the
main load is carried by the reservoir fluid. Despite
the high value of the rock pressure, the nature of
the reservoir does not provide a large margin of
safety. Apparently, this is the reason for the significant increase in productivity of wells of Tengiz
field, when reservoir drilling was accompanied by
large absorption and a significant drop in the level
of drilling mud leading to high pressure drawdown. In particular, a major accident occurred at
well No. 37 of Tengiz field in 1985. At the
opening the reservoir by just 4 m resulted in a
significant drop in the drilling mud level, which
could not be restored in time, as a result of erroneous actions of the drilling crew. The well began
to fountain with increasing flow rate, which
reached the value of 10–15 thousand tons per day
during the day. All the drilling equipment was
brought to the surface and the fountain caught fire.
The well couldn’t be shut down for a year.
Academician S. A. Khristianovich, who was
investigating the causes of the accident, wondered if the mechanism underlying these events
could be used for the good, i.e. to increase the
productivity of the well by managing the stress
state in the vicinity of the well. Testing rock
specimens from the reservoir of Tengiz field was
carried out on TILTS, which demonstrated that
imposing the stress state corresponding to pressure drawdown of about 35 MPa on rock specimens resulted in the irreversible jump-like
increase in permeability by 30–40 times. It was
93
concluded that if the pressure drawdown were
maintained at the bottom-hole for the time
required to spread the geoloosening process
inwards reservoir, the productivity of the well
could be significantly increased. Thus, the idea of
the method of geoloosening (or directional
unloading) was born.
Rocks from the reservoir of Astrakhan gas
condensate field (AGCF), which lies in geological conditions similar to those of Tengiz field,
behave in a similar way. The results of testing the
specimen from AGCF field are presented on
Figs. 6.19 and 6.20. It can be seen from the
diagrams that when stress s2 reaches the value of
140 MPa, which corresponds to a pressure
Fig. 6.19 Loading program and permeability; AGCF
field
Fig. 6.20 Deformation curves; AGKM reservoir
94
drawdown of 40 MPa, the specimen was subjected to intense inelastic deformation and a
sharp jump in permeability associated with rock
fracturing and disintegration.
A different situation is observed in reservoirs
with a high content of clay (rocks of the third
category). As noted above, these rocks start to
creep with minor pressure drawdowns, and their
permeability drops dramatically. It was impossible to initiate the fracturing process in such rocks
even with maximum pressure drawdown and the
creation of stress concentrators (perforation holes
and cuts). Thus, when testing the rock of Nizhnechutinskoye field, it was found that when even
small shear stresses corresponding to operational
pressure drawdowns were created, the permeability of the rock drops twofold compared to the
conditions of zero pressure drawdown
(Fig. 6.13). For this reason, the flow rates of
wells are significantly reduced due to formation
6
Dependence of Permeability on Stress State
of open boreholes or perforation holes in the
vicinity of the low-permeability zones. An
increase in pressure drawdown in this case only
worsens the situation and can lead to a complete
cessation of the influx.
The only possible way out in such a situation
is to unload the rocks in the bottom-hole zone
from the shear stress. In particular, for an open
wellbore it is reduced to lowering circumferential
stresses acting in its vicinity, because the maximum shear stresses in this case are determined by
the half-difference of circumferential stresses and
radial stresses equal to the fluid pressure in the
well. In practice, this can be achieved by creating
vertical cuts in the open wellbore zone before
pressure drawdown. However, the question arise:
how many cuts and of what size need to be
created in order to achieve the effective unloading of the bottom-hole zone from the circumferential stresses.
Fig. 6.21 Distribution of intensity of shear stress in the vicinity of an uncased wellbore section with two cut size of 0.1
of well radius for the conditions of Nizhne-chutinskoye field
6
Dependence of Permeability on Stress State
Mathematical modeling was carried out to
calculate stress fields in the vicinity of an open
hole with vertical slots for the conditions of the
Nizhne-chutinskoye field.
The distribution of the intensity of shear
stresses in the vicinity of an uncased wellbore
section with cuts of size of 0.1 of well radius is
shown on Fig. 6.21. Yellow and red correspond
to zones with high shear stresses and, as a result,
the decreased permeability.
It can be seen from the Fig. 6.21, that the well
is surrounded by a low-permeability “plug” with
a thickness of approximately 0.5 well radius. The
presence of cuts has a little effect, the distribution
of stresses around the well remains almost the
same as in case of the absence of the cuts.
Creating deeper vertical cuts comparable in size
with the well radius significantly changes the
situation.
95
The distribution of the intensity of shear
stresses in the vicinity of an uncased well with two
diametrically opposed vertical cuts of lengths
equal to the well radius is shown on Fig. 6.22.
It can be seen from Fig. 6.22 that the cuts
reduce the shear stresses acting along the well
contour almost twice, the zones of lowered permeability being reduced in size and moved apart
from the well contour. Thus, the presence of two
vertical cuts significantly improves situation and
maintains the permeability in bottom-hole zone.
The increase in the number of vertical cuts
does not improve the situation, but, on the contrary, worsens it. The distribution of the intensity
of shear stress around a well with four vertical
cuts of lengths equal to the well radius is shown
on Fig. 6.23.
It can be seen from the Figure that although the
rock is unloaded in the immediate vicinity of the
Fig. 6.22 Distribution of intensity of shear stress in the vicinity of an uncased wellbore section with two cuts of sizes
equal to well radius for the conditions of Nizhnechutinskoye field
96
6
Dependence of Permeability on Stress State
Fig. 6.23 Distribution of intensity of shear stress in the vicinity of an uncased wellbore section with four cuts of sizes
equal to well radius for the conditions of Nizhnechutinskoye field
well, a closed zone of reduced permeability
develops at a distance of about two radii from the
well center, which play the role of a new “plug”
that significantly reduces the flow rate into the well.
These results convincingly testify that the
stresses occurring in the bottom-hole zone can
have a significant impact on filtration properties
of the formation and, as a consequence, on the
productivity of wells. This impact can be either
positive or negative. For the conditions of a particular field (reservoir rock, conditions of occurrence, etc.) it is necessary to choose the
bottom-hole design and the value of
bottom-hole drawdowns, which provide the
maximum flow rates. This choice should be based
on conducted research of rock properties and
necessary calculations. Such an approach may
give impetus to the development of new methods
of increasing well productivity and oil recovery.
References
Karev VI, Kovalenko YuF (2013a) Triaxial loading
system as a tool for solving geotechnical problems of
oil and gas production. True triaxial testing of rocks.
CRC Press, Balkema, Leiden, pp 301–310
Karev VI, Kovalenko YuF (2013b) Well stimulation on
the basis of preliminary triaxial tests of reservoir rock.
Rock Mechanics for Resources, Energy and Environment. Proceedings of EUROCK 2013. Leiden: CRC
Press/Balkema, pp 935–940
Karev VI, Klimov DM, Kovalenko YuF, Ustinov KB
(2016) Fracture of sedimentary rocks under a complex
triaxial stress state. Mech Solids 51(5):522–526
Klimov DM, Karev VI, Kovalenko YuF (2015) Experimental study of the influence of a triaxial stress state
with unequal components on rock permeability. Mech
Solids 50(6):633–640
7
Influence of Filtration on Stress–
Strain State and Rock Fracture
in the Well Vicinity
Above, the influence of stress states on the permeability of rocks and, as a consequence, on
filtration of oil and gas in productive formations
was considered. However it is also important to
take into account the influence of filtration on the
stress state and on the size of damaged zones in
the vicinity of oil and gas wells. These questions
have been studied before (Dobrynin 1970;
Ostensen 2013; Grafutko and Nikolaevskii 1998;
Li et al. 1988; Wu et al. 2000; Zaitsev and
Mikhailov 2006; Pyatakhin 2009; Baklashov and
Kartozia 1975), but the studies were mainly
related to processing of oil fields, while the same
problems are relevant to processing of other
types of hydrocarbon fields—gas, gas condensate, highly carbonated oil, etc. which will be
referred to as a fluid hereafter.
From a mechanical point of view, these fluids
differ primarily in their density, viscosity and
dependence of these properties on pressure. The
influence of these factors on the stress–strain
state in the vicinity of the well has not been
studied sufficiently. Also, the dependence of the
size of the damaged zones appearing near the
wells on the strength characteristics of the rock
and on the relationship between the strength and
filtration parameters has not been fully
investigated.
The current study has adopted a number of
simplifications that are well justified by the
practice:
© Springer Nature Switzerland AG 2020
V. Karev et al., Geomechanics of Oil and Gas Wells,
Advances in Oil and Gas Exploration & Production,
https://doi.org/10.1007/978-3-030-26608-0_7
– the problem is solved in plane axisymmetric
statement, which is quite justified, because
the thickness of productive layers exceeds the
radius of wells by orders of magnitude;
– the permeability of rock in the damaged and
elastic regions is considered to be different.
However, inside each of the region it is
assumed to be independent of the stress state;
– the problem is considered as the stationary
one, which allows obtaining general analytical solutions; the effect of non-stationary is
considered separately.
As shown below, the solution for stationary and
non-stationary distribution of pressure is the same for
two extreme cases—when the permeability of the
rock in the damaged zone is much higher or much
lower than the permeability in the original formation.
This result provides the basis for the conclusion that,
in general, the solutions of problems in stationary
and non-stationary productions are close.
Consider the problem of stress distribution in
the vicinity of a vertical well of radius RW drilled
to a depth h in a homogeneous and isotropic
layer in the presence of a zone of radius R with
different properties around it, the deformation of
which ceased to be elastic (damaged zone). The
strength properties of the rock in the damaged
zone may differ significantly from the properties
of the original formation. The same applies to
permeability, which can be either higher or lower
than the original one (Karev and Kovalenko
2006). Further, the upper index “p” will denote
97
98
Influence of Filtration on Stress–Strain State …
7
all the characteristics of the rock in the damaged
zone. Accordingly, the upper index “e” will
indicate these values in the elastic region.
Let direct the axis z along the well axis and
introduce polar coordinates r; / in the formation
plane (the formation is considered horizontal).
Stresses rr ; r/ ; rz in the reservoir are distributed between the effective stress transmitted
through the soil skeleton sr ; s/ ; sz and the fluid
pressure p according to the law (2.20) with
aP ¼ 1, which acceptable for highly permeable
rocks.
The initial state of the oil and gas reservoirs is
considered as hydrostatic compression caused by
rock pressure q ¼ ch, where c is the average
specific weight of rocks above the formation.
r0r ¼ r0/ ¼ r0z ¼ q;
s0r ¼ s0/ ¼ s0z ¼ q þ p0
ð7:1Þ
Here p0 is the initial fluid reservoir pressure.
The difference between the pressure p and the
initial formation pressure p0 will be used hereafter, i.e.
p0 ¼ p p 0
ð7:2Þ
ser ¼ ke þ 2ler
se/ ¼ ke þ 2le/
e ¼ er þ e/
where k; l—Lamé constant. Strains er ; e/ in the
radial and circumferential directions are expressed through the radial displacement u by means
of Cauchy relations
er ¼
du
u
du u
; e/ ¼ ; e ¼ er þ e/ ¼
þ ð7:6Þ
dr
r
dr
r
Sequential substitution of the Cauchy relations (7.6) into Hooke’s Law (7.5) and equilibrium Eq. (7.4) gives Lamé equation in polar
coordinates
d du u
dp0
ðk þ 2lÞ
þ
¼
dr dr
r
dr
rr ¼ sr p0 p0 ;
r/ ¼ s/ p0 p0 ;
rz ¼ sz p0 p0
ð7:3Þ
ser ¼ C dsr
sr s/ dp0
þ
¼
dr
r
dr
D
1 2m I ðr Þ
þ p0 ð r Þ r2
1 m r2
ð7:4Þ
Outside the damaged zone, the stress related
to strains by Hooke’s law
ð7:8Þ
D
m
1 2m I ðr Þ
p0 ð r Þ þ
þ
ð7:9Þ
2
r
1m
1 m r2
where C и D are constants of integration, and
Zr
I ðr Þ ¼
Since the reservoir thickness is many times
the diameter of the well, it can be assumed that
the reservoir after drilling is subjected to plane
strain conditions. In this case, the equilibrium
equation is written in the form
ð7:7Þ
the general solution of which for the arbitrary
distribution of p0 ðr Þ results in the following distribution of stresses
se/ ¼ C þ
Hence
ð7:5Þ
r 0 p0 ðr 0 Þdr 0
ð7:10Þ
R
The obtained expressions for the stresses
acting in the elastic zone are valid for arbitrary
distribution of pore pressure, including those
corresponding to unsteady flaws.
The stresses acting in the damaged zone are
obtained from the equilibrium condition (7.4)
and a fracture criterion, i.e. Mohr-Coulomn criterion (1.23), which for the considered conditions
of axial symmetry can be written as (Zhuravlev
et al. 2014).
7
Influence of Filtration on Stress–Strain State …
spr sp/ ¼ ap ðspr H p Þ H p ¼ sp0 cotqp0
ap ¼
2 sin qp0
1 sin qp0
ð7:11Þ
Here sp0 ; qp0 are the cohesion and the internal
friction angle of the medium in the damaged zone.
System of Eqs. (7.4), (7.11) for finding
stresses spr ; sp/ is statically determinate, and its
solution satisfying the boundary condition on the
well contour
spr ðRw Þ ¼ 0
ð7:12Þ
99
fulfillment of the ultimate equilibrium condition
for stresses ser and se/ similar to (7.11), i.e.
ser ðR Þ se/ ðR Þ ¼ ae ser ðR Þ H e ; ae
2 sin qe0
¼
;
1 sin qe0
H e ¼ se0 ctgqe0
Here se0 ; qe0 are adhesion and the internal
friction angle of the medium in the elastic zone.
– boundary condition on the outer contour
Is written as follows
spr ðr Þ
I 2 ð r Þ ¼ ð R Þ a
p
Zr
Rw
ser ðRk Þ ¼ q þ p0
¼ I 2 ðr Þ
ð7:13Þ
0
p
1 dpðr Þ a H
0 dr 0
r 0ap dr 0
r
p
ð7:14Þ
The circumferential stress sp/ is found from
Eq. (7.11)
sp/ ¼ ð1 þ ap Þspr ap H p
ð7:15Þ
Thus, inside ðr\R Þ and outside ðr [ R Þ the
damaged zone, the stress state is determined by
(7.13), (7.14), (7.15), and by (7.8)–(7.10),
respectively. Three constants C; D; R involved
in these expressions are determined from the
boundary conditions:
– two conditions on the boundary of the damaged zone r ¼ R , namely
continuity of radial stress
ser ðR Þ ¼ spr ðR Þ
ð7:16Þ
ð7:17Þ
ð7:18Þ
Here Rk is the radius of the external reservoir
boundary, where the pressure is supposed to be
equal to the formation pressure p0 ; ke is permeability in the elastic zone; kp is permeability in
the damaged zone.
Consider particular practical cases.
Stress State and Size of Damaged Zone in
the Vicinity of the Well in the Absence of
Filtration
For in the well pw equal to the initial pressure p0
no pressure gradient occur, and in equilibrium
(7.4) the right part disappears. Condition
spr ðRw Þ ¼ 0 is followed by
"
spr
¼H
p
sph
¼H
p
"
r
1
Rw
ap #
p#
1 þ sin qp0 r a
1
1 sin qp0 Rw
ð7:19Þ
Conditions on the external reservoir boundary
(7.18) can be attributed to infinity
100
7
ser ð1Þ ¼ q þ p0
ð7:20Þ
Influence of Filtration on Stress–Strain State …
account that p Rk ¼ p0 , leads to the following
expression for radial stress at point R
0
Then
2
R
þ q þ pw
r
2
R
se/ ¼ ðq þ pw H e Þ sin qe0
þ q þ pw
r
ser ¼ ðq þ pw H e Þ sin qe0
ð7:21Þ
and for the radius of the disturbed zone R , we
have
p
R
q þ pw ðq þ pw H e Þ sin qe0 1=a
¼ 1
Rw
Hp
ser ðRÞ
¼@
ae
1
ðlnðR =Rw Þ lnðRk =Rw ÞÞ
þ 2ð1mÞ
1 sin qe0 þ se0 cos qe0
pðrÞ ¼
pw þ aP1
aP1 ¼
dp aP1
¼
dr
r
r
ln ;
Rw
Dpw
kP ðk1P ln RRw þ
1
ke
ln RRk Þ
Using condition (7.16), (7.17) and (7.27), we
obtain the equation to determine R
"
H1p
R
1
Rw
a p #
0
¼@
ae1 ¼
ke ðk1p
dp ae1
¼
dr
r
Rk
;
r
Dpw
þ
ln RRw
1
ke
ln RRj Þ
1
12m e
a1
q þ p0 4ð1mÞ
ae
1
þ 2ð1mÞ
ðlogðR =Rw Þ logðRk =Rw ÞÞ
1 sin qe0 þ se0 cos qe0
A
ð7:28Þ
If the permeability in the damaged zone is
significantly higher than the original one, i.e.
kp =ke 1 from (7.23), (7.24) and (7.28) we find
(neglecting term
ð12mÞ
2 log Rk =R
1)
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
p0 pw
u
1 sin qe0 se0 cos qe0
ap
t
2ð1mÞ q p0
R
¼
1þ
Rw
sp0 cotqp0
ð7:29Þ
ð7:23Þ
ð7:24Þ
where Dpw ¼ p0 pw is draw down in the well.
fort r [ R (in the elastic zone)
pðrÞ ¼ p0 ae1 ln
A
ð7:27Þ
ð7:22Þ
Accounting for the Effects of Filtration of
Uncompressible Fluid
For distribution of fluid pressure in the reservoir
in the presence of two zones of different permeability in a steady flow we have (Leibenzon
1947):
for r\R (in the damaged zone)
1
12m e
a1 þ
q þ p0 4ð1mÞ
ð7:25Þ
ð7:26Þ
Neglecting a term with coefficient R2 =R2k
(which is justified for Rk R), and taking into
For another limiting case, when the permeability in the damaged zone is much lower than
the permeability in the elastic region, i.e. for
kp =ke 1, from (7.23), (7.24) and (7.28) we
obtain
ffi
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
e se cos qe
ð
q
p
Þ
1
sin
q
R
u
0
p
0
0
0
a
¼ t
1þ
ð1sin qp0 Þ
p0 pw
Rw
sp0 cotqp0 logðR
p
=R Þ
2 sin q
w
0
ð7:30Þ
The results of calculations of the size of the
damaged zone in the vicinity of the well for
various parameters are presented below. For all
cases it was assumed m = 0.2, q = −90 MPa,
p0 = 60 MPa, ln Rk =Rw = 8.
7
Influence of Filtration on Stress–Strain State …
Fig. 7.1 Dependence of the radius of damaged zone on
pressure drawdown; se0 ¼ sp0 ¼ 10 MPa, qe0 ¼ qp0 ¼ 30 ;
(1) kp/ke = 1; (2) kp =ke = 0.2; (3) kp =ke = ∞
101
Fig. 7.3 Dependence of the radius of damaged zone on
pressure drawdown; sp0 = 3 MPa, se0 = 10 MPa, qp0 = 20°;
qe0 = 30°; (1) kp =ke = 1; (2) kp =ke = 5; (3) kp =ke = ∞
For this case it is also assumed that the
strength properties in the damaged and elastic
zones coincide.
Fig. 7.2 Dependence of the radius of damaged zone on
pressure drawdown; sp0 = 5 MPa, se0 = 10 MPa, qp0 = 20°;
qe0 = 30°; (1) kp =ke = 1; (2) kp =ke = 5; (3) kp =ke = ∞
Figure 7.1 depicts the dependence of the
damaged zone radius on pressure draw down for
the case of the lower permeability in the damaged zone comparing to permeability in the
elastic zone. Here it was assumed that the
strength parameters in both zones coincide. This
assumption is justified by the observation that the
reduction of permeability usually occurs before
the beginning of fracture (Karev and Kovalenko
2006), and therefore is not accompanied by a
significant change in strength properties.
Figures 7.2 and 7.3 depict the dependence of
the damaged zone radius on pressure draw down
for the opposite case, corresponding to higher
permeability in the damaged zone comparing to
permeability in the elastic zone.
Influence of Dependence of Fluid Compressibility and its Viscosity on Pressure
The above study was carried out for an incompressible fluid, which can be as an approximation
applied to oil. The case of compressible fluid
(gas, gas condensate, etc.) is important for the
practice too. Accounting for dependence of fluid
density and viscosity on pressure results in
nonlinearity of the equations, which complicates
solving. To simplify the analysis, we will assume
that the permeability of the layer is the same for
the elastic and damaged zones. The filtration law
preserves the form of Darcy law
q f ¼ k
c ð pÞ
grad p
lð p Þ
ð7:31Þ
Here q f is mass flow of fluid; k is permeability
of the rock; cð pÞ is the fluid density; lð pÞ is the
fluid viscosity; p is pressure.
For the fluid flow, the equation of compatibility remains valid, for the stationary case it has
the form
div q f ¼ 0
ð7:32Þ
102
7
Influence of Filtration on Stress–Strain State …
In the two-dimensional axisymmetric case
Eqs. (7.32) and (7.31) followed by
k
c ð pÞ
grad p ¼ C0 =r
lð pÞ
ð7:33Þ
where C0 is a constant of integration.
The simplest variant of density dependence on
pressure is the linear law
c ð pÞ ¼ A c þ Bc p
ð7:34Þ
Similarly, for viscosity dependence on pressure the linear law is also may be accepted
l ð p Þ ¼ Al þ Bl p
ð7:35Þ
Although the viscosity of ideal gas in a wide
range of pressures does not depend on pressure
(for example, Landau and Lifshitz 1976), for real
gases an increase in viscosity with pressure is
observed (for example, Golubev 1959). If the
increase of viscosity is approximately the same
as the increase of density, the distribution of
pressure of the compressible fluid would be the
same as for the incompressible fluid.
Consider, however, a more general case.
Since both density and viscosity depend on
pressure only (within the framework of the considered model), the dependence for the combination in question may be written as follows
c ð pÞ
00
0 p
k
¼ A 1þB
lð pÞ
p0
ð7:36Þ
Here the coefficients in the right-hand side of
the equation are considered to be known
parameters of the model. Let refer coefficient B0
as a generalized parameter of compressibility.
The distribution of pressure for the stationary
problem under consideration will not directly
depend on coefficient A00 . Coefficient B0 is possible to express through physical parameters density and viscosity at atmospheric (zero)
pressure cA ; lA and at reservoir pressure c0 ; l0
B0 ¼
c0 =l0 cA =lA
c0 =l0
ð7:37Þ
If there is no dependence of viscosity on
pressure, the formula (7.37) is simplified as
following
B0 ¼
c0 cA
c0
ð7:38Þ
Substituting (7.36) into (7.31) and solving the
resulting equation with boundary conditions
pðRw Þ ¼ p0 Dpw , pðRk Þ ¼ p0 , we get
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
2
r
p0
0 þ C ln
0
þ
C
pð r Þ ¼ p0
0
B02 B0 p0
Rc
B
ð7:39Þ
Here, the constants are
C 0 ¼ ðp0 Dpw Þ þ
B0
ðp0 Dpw Þ2
2p0
B0 2
p
2p0 w
Dpw
B0
Dpw
C0 ¼ Rk 1 þ
p0 p0
2
ln Rw
0
Dpw
B
Dpw
pw þ
¼ Rk 1 þ
p0
2
ln R
ð7:40Þ
¼ pw þ
ð7:41Þ
w
For distribution of pressure (7.39) expression
(7.10), (7.14) are converted as follows
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
B0
B0
r
1 þ 2C 0 þ 2C0 ln
p0
p 0 Rw
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2w p0
B0 r 2 R2w
1 þ 2C 0 p0
0
2B
p0
2B0
rffiffiffiffiffiffiffiffiffiffiffiffiffi
1 pp0 C0 2
p0 þ 2B0 C 0
exp
R
w
4
B0
B0 C0
ffi
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p0
2C0
r
erfi
þ
þ 2 ln
Rw
B0 C0
C0
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0
p0
2C
erfi
þ
0
B C0
C0
r 2 p0
I ðr Þ ¼
2B0
ð7:42Þ
7
Influence of Filtration on Stress–Strain State …
Fig. 7.4 Dependence of the radius of damaged zone on
pressure draw down for qe0 ¼ qp0 ¼ 30 and B0 = 0.1:
(1)
se0 ¼ sp0 ¼ 10 MPa;
(2)
se0 ¼ sp0 ¼ 5 MPa;
p
e
(3) s0 ¼ s0 ¼ 3 MPa
103
Fig. 7.6 Dependence of the radius of damaged zone on
pressure draw down qe0 ¼ qp0 ¼ 30 and B0 = 0.7.
(1)
se0 ¼ sp0 ¼ 10 MPa;
(2)
se0 ¼ sp0 ¼ 5 MPa;
p
e
(3) s0 ¼ s0 ¼ 3 MPa
taking into account that pðRk Þ ¼ p0 , conditions
(7.16)–(7.18), lead to equation for radius of the
damaged zone R
qþ
1 2m I Rk
pðR Þ 1 sin qe0 þ H e sin qe0 ¼ I2
þ
2
1 m Rk
2ð1 mÞ
ð7:44Þ
Fig. 7.5 Dependence of the radius of damaged zone on
and
B0 = 0.5:
pressure
draw
qe0 ¼ qp0 ¼ 30
(1)
se0 ¼ sp0 ¼ 10 MPa;
(2)
se0 ¼ sp0 ¼ 5 MPa;
(3) se0 ¼ sp0 ¼ 3 MPa
"
p #
a
R
I2 ¼ H p
1
Rw
rffiffiffiffiffiffiffiffiffiffiffiffiffi
p
ap
pp0 C0
a p0
ap C 0
R
þ
þ
exp
p
0
0
2a B
2B C0
C0
Rw
" sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p0
C0
R
erf ap
þ
þ ln
0
2B C0 C0
Rw
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
#
p0
C0
erf ap
þ
2B0 C0 C0
ð7:43Þ
By neglecting the term with coefficient
R2 =R2k (which is justified for Rk R) and
Figures 7.4, 7.5, and 7.6 depicts the dependence of the radius of damaged zone on pressure
draw down for various values of compressibility
B0 . The curves in the figures correspond to various cohesions.
Influence of Unsteady Fluid Flow
The distribution of fluid pressure in the damaged
zone, accounting for its relatively small size and
increased permeability, can still be considered as
stable (Leibenzon 1947)
p0 ¼ p0w þ a2 log
a2 ¼
r
;
Rw
dp0 a2
¼
dr
r
Dp0w
logðR =Rw Þ
ð7:45Þ
ð7:46Þ
Here
p0w ¼ pw p0 ; Dp0w ¼ p0 p0w
¼ p pw [ 0; p0 ¼ p p0
ð7:47Þ
In the damaged zone the effective stresses are
related by (7.11). From the equation of
104
7
equilibrium (7.4) and (7.45), with account for
that spr ðRw Þ ¼ 0 we obtain
"
spr
r
¼ H2 1 Rw
ap #
;
H2 ¼ H p a2
ap
ð7:48Þ
Expressions (7.8) and (7.9) for the stresses
acting in the elastic zone remain valid for arbitrary pore pressure distributions in this area,
including those corresponding to unsteady flaws,
the particular distribution of pressure having no
influence on constants C and D, but affecting
integral IðrÞ in (7.10).
For any non-stationary process, according to
(7.4), p0 ðr Þ ! 0 as r ! 1 and p0 ðr Þ ! 0 as
1=r n , where n [ 0. If this condition is satisfied,
then lim I ðr Þ=r 2 ¼ 0. Besides, expression (7.10)
r!1
is followed by I ðRÞ ¼ 0.
This means that the stresses at the point R do
not depend on a particular distribution of pressure but are determined by the pressure p at that
point only.
Constants of integration C and D, included in
the expressions for stresses (7.8), (7.9), are found
by substituting them into the conditions (7.20),
(7.17), and then from (7.8) for the radial stress at
the point R we obtain
ser ðRÞ ¼
q þ p0 þ
p0
2ð1 mÞ
e
1 sin q0 þ se0 cos qe0
ð7:49Þ
Using condition (7.16) and relations (7.48),
(7.49), we obtain equation for R
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
p0
u
ap
1 sin qe0 þ se0 cos qe0
Q þ p0 þ 2ð1mÞ
t
R
¼
1
Rc
H2
Influence of Filtration on Stress–Strain State …
It follows from (7.50) and (7.51) that in order
to determine the radius of the damaged zone R it
is sufficient to know the value of fluid pressure p
at its boundary r ¼ R , which is to be obtained
from the solution of the filtration problem.
However, for two practically important cases the
solution can be written out immediately.
1. If the permeability in the damaged zone is
significantly higher than the permeability in
elastic zone, i.e., kp =ke 1, then from
(7.23), (7.24), (7.50) and (7.51) taking into
account (7.47) we have
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
p0 pw
u
1 sin qe0 se0 cos qe0
ap
t
2ð1mÞ q p0
R
¼
1þ
sp0 cotqp0
Rw
ð7:52Þ
2. If the permeability in the damaged zone is
low compared to the permeability in elastic
zone, i.e., kp =ke 1, then from (7.23),
(7.24), (7.50) and (7.51) taking into account
(7.47) we find
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
u
e se cos qe
ð
q
p
Þ
1
sin
q
R
u
0
p
0
0
0
a
¼ t
1þ
ð1sin qp0 Þ
p0 pw
Rw
sp0 cotqp0 logðR
p
=R Þ
2 sin q
w
0
ð7:53Þ
Expressions (7.52) and (7.53) coincide exactly
with expressions (7.29) and (7.30) for the case of
a stationary flow in the elastic zone. This fact is
the basis for the conclusion that, in general, the
solution of problems in stationary and
non-stationary productions will be close.
The following conclusions can be made from
the analysis.
ð7:50Þ
where due to (7.47)
H2 ¼ sp0 cotqp0 p pw
ð1 sin qp0 Þ
2 sin qp0
logðR =Rw Þ
ð7:51Þ
1. The size of the damage zone depends essentially on the strength properties of rock in the
elastic and damaged zones. A decrease in
strength parameters of the rock in the damaged zone leads to a noticeable increase in its
radius.
7
Influence of Filtration on Stress–Strain State …
2. The presence of fluid filtration from the layer
into the well increases the size of the damaged
zone, and with increase in the pressure gradient, the size of the damaged zone increases.
3. The size of the damaged zone is significantly
influenced by the ratio of permeability in the
damaged and elastic zones. The growth of
permeability in the damaged zone leads to a
decrease in its size, and decrease in permeability in the damaged zone leads to its
expansion.
4. The radius of the damaged zone increases
with the increase in generalized compressibility parameter B0 . Thus, the more compressible is fluids, e.g. gas, the more
pronounced is expansion of the damaged
zone. This result becomes especially important for reservoirs, composed by relatively
weak rocks with low cohesion.
5. In the case of a non-stationary flow, stresses
at the boundary of the damaged zone and, as a
consequence, the size of the damaged zone,
do not depend on a particular distribution of
pressure in the reservoir, but is determined
solely by the fluid pressure at that boundary.
The solutions of the non-stationary problem
for two limiting cases of ratios of permeability in the damaged and elastic zones
obtained on the basis of this conclusion
coincided exactly with the analogous solutions for the stationary problems. This
observation allows us to suppose that, in
general, the solutions of problems in stationary and non-stationary approximations are
close.
105
References
Baklashov IV, Kartozia BA (1975) Rock mechanics (in
Russian). M.: Nedra, 271p
Dobrynin VM (1970) Deformations and changes in
physical properties of oil and gas reservoirs (in
Russian). M.: Nedra, 239p
Golubev IF (1959) Viscosity of gases and gas mixtures (in
Russian). M.: Fizmatlit, 377p
Grafutko SB, Nikolaevskii VN (1998) Problem of the
sand production in a producing well (in Russian).
Fluid Dyn 33(5):745–752
Karev VI, Kovalenko YuF (2006) Dependence of the
bottom-hole formation zone permeability on the
pressure drawdown and the bottom-hole design for
different types of rocks (in Russian). In: Technologies
of the fuel and energy complex, 6:59–63
Li YD, Rabbel W, Wang R (1988) Poro-elastic response
of a borehole in a nonhydrostatic stress field. Int J
Rock Mech Min Sci Geomech Abstr 25(3):171–182
Leibenzon LS (1947) Movement of natural liquids and
gases in porous medium (in Russian). M.-L.:
Gostekhizdat, 244p
Landau LD, Lifshitz EM (1976) Statistical physics (in
Russian). Part 1: Edition 3, supplemented. M.:
Science, 584p
Ostensen RW (2013) The effect of stress-dependent
permeability on gas production and well testing. SPE
Formation Evaluation 1(3):227–235
Pyatakhin MV (2009) Critical flow rate of rock destruction in the bottomhole zone of a horizontal well (in
Russian). Gas Industry 7:27–33
Wu YS, Pruess K, Witherspoon PA (2000) Integral
solutions for transient fluid flow through a porous
medium with pressure-dependent permeability. Int J
Rock Mech Min Sci 37:51–61
Zaitsev МV, Mikhailov NN (2006) Effect of residual oil
saturation on the flow through a porous medium in the
neighborhood of an injection well. J Fluid Dyn 6:568–573
Zhuravlev AB, Karev VI, Kovalenko YuF, Ustinov KB
(2014) The effect of seepage on the stress–strain state
of rock near a borehole. J Appl Math Mech 78
(1):56–64
8
Results of Tests of Rock Specimens
by Using TILTS
Theoretical and experimental studies performed in
Laboratory of Geomechanics of Ishlinsky Institute
for Problems in Mechanics RAS have demonstrated that the geomechanical approach may
serve as the basis for solving the most important
problems arising during the development of oil
and gas fields: ensuring the stability of wellbores
during drilling and operation, increasing the productivity of oil and gas wells, and increasing oil
and gas recovery from reservoirs (Karev et al.
2015, 2016a, b, c, 2017a, b, 2018a, b, c, d, e;
Kovalenko et al. 2016; Klimov et al. 2015; Zhuravlev et al. 2012; Karev and Kovalenko 2006,
2013). To solve these problems, a new approach
was proposed based on the direct modeling of the
processes of deformation and failure of rock in the
well vicinity and their influence on the filtration
properties of rocks using the Test System of
Three-axis Independent Loading (TILTS) (Karev
et al. 2015, 2017a; Kovalenko et al. 2016).
In implementing the developed approach to
solving particular practical problems, the
obtained results and the developed loading programs for rock specimens are used.
The results of physical modeling of real
mining situations using TILTS on rock specimens from various oil and gas fields are presented below.
© Springer Nature Switzerland AG 2020
V. Karev et al., Geomechanics of Oil and Gas Wells,
Advances in Oil and Gas Exploration & Production,
https://doi.org/10.1007/978-3-030-26608-0_8
8.1
Results of Physical Modeling
of Resistance to Failure
of Inclined and Horizontal Wells
for Particular Objects
Results of Physical Modeling of Inclined
Wells for Particular Object
Prior to any testing by using TILTS, the velocities of longitudinal waves were measured in
cubic specimens along all three specimen’s axes.
For this purpose a specialized installation had
been designed. The installation consists of two
ultrasonic wave sensors, between which the
specimen under study is installed. To visualize the measurement results, the electrical signals from both sensors are displayed at the
oscilloscope. Ultrasonic waves of 1.25 MHz
frequency are passed between the sensors and the
time of the waves passing through the specimen
and damping of the amplitude are measured. The
velocities are measured in three directions: the
axis 1 coinciding with the core axis, and the axes
2 and 3 in two mutually perpendicular directions
in the horizontal plane. The used frequency
corresponds to wave length about 5 mm. The
scheme is presented on Fig. 8.1.
107
108
Fig. 8.1 Scheme of installation for determining the
p-wave propagation velocity in the rock; 1—emitter, 2
—receiver, 3—specimen, 4—oscilloscope, 5—pulse
generator
Based on the results of measurement, the
conclusion is made about the degree of anisotropy of the rock to be tested.
If the rock is isotropic (the p-wave velocities in
the three axes of the specimen are close to each
other), the specimens are tested in accordance with
program presented at Fig. 5.9, which corresponds to an isotropic impermeable environment.
If the rock is anisotropic (transversally isotropic), the p-wave velocities in the horizontal
plane are close to each other and noticeably
lower in the vertical axis of the core. In this case,
for determinating of permissible mud densities
that do not lead to the well failure for various
angles of inclination to the vertical, the direct
physical simulation of deformation and fracture
processes in the vicinity of the well is used.
In experiments, the specimens are subjected to
stress states as close as possible to the stress state
occurring on the contour of an inclined well at
various angles of inclination and pressures on its
bottom-hole. Each angle of inclination of the
well corresponds to a specimen made cut at
exactly the same angle from the vertical. Specimens are cut so that the angle between their
vertical axis and the core axis corresponds to the
inclination angle of the well; usually the angles
are 0 ; 15 ; 30 ; 45 ; 60 ; 75 . The specimens are
then placed in TILTS and loaded according to the
program shown at Fig. 5.9 for impermeable
rocks.
8
Results of Tests of Rock Specimens by Using TILTS
The specimen is held for a long period of time
under the constant load corresponding to the
bottom-hole pressures within the range of interest
to record creep deformation. In the course of
each experiment, the strains are recorded in each
of the three directions.
Such experiments are carried out for various
bottom-hole pressures. If during keeping the
specimen under the constant stresses its deformation stops increasing after some time, the
bottom-hole pressure corresponding to the acting
stresses is considered acceptable. If the deformation increase does not stop over time, then the
stability of the wellbore should be expected to be
lost at the bottom-hole pressure corresponding to
the acting stresses.
As a result of the analysis of these experiments, permissible parameters of well drilling
(inclination angle, drilling mud density, wellbore
stability time) and permissible pressure drawdowns during well drilling in productive formations and host rocks are determined.
Results of Physical Simulation of Inclined
Wells for Particular Fields
Ulyanovskoye field of PJSC Surgutneftegas
As an example of application of the developed
methodology for determination of the critical
well inclination angles for a particular field, the
results of research for the Ulyanovskoye field of
PJSC “Surgutneftegas” are given.
For the direct modeling of the conditions
occurring on the contour of inclined wells and for
studying the dependence of the ability of rocks to
withstand the acting stresses on the inclination
angles of the wells, core material from a number
of vertical wells of Ulyanovskoye field was tested by using the TILTS. Rocks from overlying
formations have been studied.
Core specimens in the form of a cube with the
rib of 50 mm were prepared, corresponding to
the angles of inclination of the well to the vertical
of 0°, 15°, 30°, 45°, 60°. Figure 8.2 depicts a
rock specimen cut at an angle of 30° to the
vertical. Layers, in the plane of which the specimen was destroyed are clearly visible. As the
core material was taken from overburden rocks,
8.1 Results of Physical Modeling of Resistance …
109
Fig. 8.4 Dependence of creep strains on the well inclination; mud density 1.12 g/cm3
Fig. 8.2 Specimen, made cut at 30° to the vertical (after
test)
Fig. 8.3 Dependence of creep strains on the well inclination; mud density 1.2 g/cm3
the specimens were tested according to the
loading program for impermeable rocks. Figure 8.3 depicts rock creep diagrams for five
specimens made cut at 0°, 15°, 30°, 45°, 60°.
When the radial stress s3 reached the value
corresponding to the mud density of 1.2 g/cm3,
further increase of loading was stopped and the
strains were measured over time. The diagram
shows the strain along the third axis versus time
in minutes.
The diagram demonstrates that the specimen
made cut at angle 0° manifested no creep, the
specimens made cut at 15° and 30° manifested
some restricted creep, the specimens made cut at
45° and 60° start to creep with increasing strain
rate up to failure.
Figure 8.4 depicts creep diagrams for specimens made cut at angles of 0°, 15°, 30°, 45°, 60°
for the radial stress corresponding to mud density
of 1.12 g/cm3.
It can be seen from the plot that, with greater
the pressure drawdown at the bottomhole, already at lower angles of inclination specimens
started to creep with increasing speed, followed
by collapse. For mud density 1.12 g/cm3 the
specimens started to failure from 30°.
These results are consistent with practice.
Drilling mud with density of 1.2 g/cm3 is used
when approaching the roof of the reservoir. When
the angle of inclination of the well reaches 45°–
50°, the wellbore stability is often lost. The studies
have shown that when using less dense drilling
fluids (1.12 g/cm3), the loss of stability can also
occur at smaller angles (30°). Therefore, in order
to ensure safeness, it is necessary to use heavier
mud, with a density of more than 1.2 g/cm3, when
reaching angles of 45°–50°.
Fedorovskoye Field
According to the above scheme, 4 specimens
from the Fedorovskoye field reservoir were tested, selected from the interval of 2728.68 to
2791 m. Since the core diameter was about
80 mm, the specimens were made in the form of
40 mm cubes. The specimens were made cut at
angles of 0°, 30°, 45°, 60° to the core axis, which
corresponds to various points on the contour of
the horizontal well, Fig. 3.8. The specimens were
110
8
marked as follows: axis 1 of the specimen coincided with the core axis, the orientation of axes 2
and 3 was (Kovalenko et al. 2016; Karev et al.
2017a, 2018f).
Since the rocks under study were selected
from the reservoir, the loading program shown in
Fig. 5.8 was used. At the first stage, the specimens were loaded with equi-component compression up to 36 MPa, corresponding to the
in situ effective stresses (point A in Fig. 5.8).
Then the stress states occurring in the vicinity
of the horizontal well for various values of
bottom-hole pressure at a given depth, i.e. at
different values of drilling mud density (AB
sections) were simulated. Each point on AV
section corresponded to a certain value of
bottom-hole pressure and to a certain value of
drilling mud density.
The specimen was held for rather long time
under the constant load corresponding to the mud
density within the range of 1.40–1.0 g/cm3, to
register creep deformation.
If the specimen was not failed during modeling pressure drawdown into the well, the experiment was continued and process of further
drawdown was simulated. The specimen was
held for sufficiently long time under constant
load corresponding pressure drawdowns of 0.5,
1.0, 1.5 MPa, etc. to register creep deformation.
The strains of the specimen measured in three
directions during the test are shown in Figs. 8.5,
8.6, 8.7 and 8.8. In each figure, the first plot
depicts the specimen loading program, the second one depicts the stress-strain curves, and the
third one depicts the creep curves for various
bottom-hole pressure drawdowns dp.
Specimen F-2
Specimen F-1
The core depth was 2727.68 m. The specimen
was cut at angle 0 to the core axis, which corresponds to point M on the horizontal well contour, Fig. 3.8. Test results are shown in Fig. 8.5.
Testing of the specimen reviled its high strength
—the creep started noticeably only on the
drawdown of 10 MPa, and drawdown of 11 MPa
caused failure, Fig. 8.5c.
Results of Tests of Rock Specimens by Using TILTS
The specimen had been taken from depth of
2730.5 m, and was cut at an angle of 30° to the
core axis, which corresponds to angle u ¼ 30 in
Fig. 3.8. The test results are shown in Fig. 8.6.
Figure 8.6c shows that at any mud density
higher than 1.0 g/cm3 the creep of the specimen
was insignificant. However, starting with drawdown of 1.0 MPa, the specimen deformed significantly, starting with drawdown of 1.5 MPa,
the specimen creep rate increased, and at drawdown of 2.0 MPa creep became very pronounced. Starting with drawdown of 4.0 MPa,
the creep became unrestricted, i.e., its rate has
increased with time. Therefore, it can be assumed
that the maximum permissible pressure drawdown is 1.5–2.0 MPa.
Specimen F-4
The depth of sampling is 2735.2 m. The specimen was cut at an angle of 45° to the core axis,
which corresponds to angle u ¼ 45 in Fig. 3.8.
Test results are shown in Fig. 8.7. From
Fig. 8.7c it can be seen that the specimen
deformation was insignificant at drilling mud
density above 1.0 g/cm3. However, while modeling the drawdown of 0.5–1.0 MPa the specimen started to creep rather intensively and then
failed under conditions corresponding to drawdown of 2.5 MPa, Fig. 8.7c. Therefore, pressure
drawdown can be considered as acceptable up to
0.5–1.0 MPa.
Specimen F-7
The specimen was selected from depth of
2791 m, the angle of inclination from the core
axis u ¼ 60 (Fig. 3.8).
Test results are shown in Fig. 8.8. This specimen turned out to be much less durable than the
rest of the specimens from the Fedorovskoye field.
As it follows from Fig. 8.8c, the specimen began
to creep noticeably under conditions corresponding the drilling mud density of 1.2 g/cm3, and for
density of 1.1 g/cm3 the creep became unlimited.
8.1 Results of Physical Modeling of Resistance …
111
Fig. 8.5 Results of test of specimen F-1cut at h ¼ 0
Below the results of the tests of rock specimens from the Fedorovskoye field reservoir are
described.
Table 8.1 depicts the results of measuring of
p-wave velocities in three axes in two specimens
made cut at zero angle to the vertical. Axis 1
coincides with the vertical axis of the core, and
the axes 2 and 3 lie in the horizontal plane.
Specimen F-1 was taken from a much shallower
depth than specimen F-8.
It is seen from Table 8.1 that p-wave velocity
in specimen F-1 along the vertical direction is
lower than that in the horizontal plane, but
insignificantly. However, in the specimen F-8,
selected from a much greater depth, the velocity
along the core axis is significantly lower than in
the perpendicular directions.
This suggests that at depth of 2791 m the rock
is much more anisotropic than at depth of
2727.68 m, i.e. anisotropy increases with the
depth, and the strength decreases accordingly.
This is also confirmed by the test data of the
specimens by using TILTS.
The test results given above are followed by
the conclusion that at depth of 2727.68 m
(specimen F-1) drilling under pressure drawdown
of more than 3.5 MPa is possible; for depth of
2730 m (specimen F-2) the acceptable diapason
of pressure drawdown is 1.5–2.0 MPa, at depth
of 2735.2 m (specimen F-3) the acceptable
112
8
Results of Tests of Rock Specimens by Using TILTS
Fig. 8.6 Results of test of specimen F-2 cut at h ¼ 30
diapason of pressure drawdown is 0.5–1.0 MPa,
and at depth of 2791 m (specimen F-7) drilling is
possible only under a lower magnitude of pressure drawdown.
From the above test results it can be concluded
that drilling horizontal wells under a drawdown in
the Fedorovskoye reservoir is associated with
high risks of loss of wellbore stability.
Filanovsky Field
It was noted above that the anisotropy of deformation and especially strength properties of
rocks can have a decisive influence on the result
and prediction of deformations and failure of
well walls. It may happen that elastically isotropic rocks manifest pronounced strength anisotropy. This is illustrated below by experiments
on core material from the Filanovsky field
(Karev et al. 2016a, 2017b).
The field is located on the Caspian Sea
shelf. According to the project the field is to be
developed by means of long horizontal wells.
The difficulty lies in the fact that the productive strata are composed mainly by rocks of
five lithotypes, which differ significantly in
their deformation, strength and filtration
properties. Therefore, the issues arise of
determining the critical flow rate and drawdown, the excess of which would lead to well
destruction, and of studying the influence of
drawdown on filtration properties of the
lithotype groups.
To answer to these questions, a series of tests
of rock specimens from the reservoir of the
8.1 Results of Physical Modeling of Resistance …
113
Fig. 8.7 Test results of the F-4 specimen cut at h ¼ 45
Filanovsky field was carried out by using the
TILTS.
The core material of five lithotypes from
exploration wells No. 2, 4, and 5—sandstones,
siltstones, clayey sandstones, gravelites and
interstratifications were used for testing. Cubic
specimens of 40 mm were cut from the core,
one of the faces being perpendicular to the core
axis.
Prior to the tests the velocities of the longitudinal elastic waves along each of the three axes
of the specimens were measured. Specimens of
all lithotypes manifested very low discrepancy in
the velocities in each direction, which indicates
isotropy of the elastic properties of the studied
rocks.
Below are the results of tests of specimens
taken from the reservoir of the Filanovsky field.
The test program corresponds to the change of
stresses on contour of the horizontal well under
decreasing bottom-hole pressure. The program is
shown in Fig. 5.8.
Figure 3.8 shows the circumferential and
radial stress r/ ; rr acting on the horizontal well
contour at two points M and N. The used loading
programs corresponded to these two points. The
key difference when testing specimens for M and
N points is that the maximum compressive
114
8
Results of Tests of Rock Specimens by Using TILTS
Fig. 8.8 Results of tests of specimen F-7 cut at h ¼ 60
Table 8.1 Velocity of
longitudinal waves of
specimens from the
reservoir of Fedorovskoe
field
№
№ specimen
Depth, m
5
F-1
2735.2
5
F-8
stresses r/ at point M acts normally to the core
axis and at point N it acts parallel to it.
Accordingly, rock specimens were placed into
the loading unit of the TILTS.
2791
№ axis
Velocity,
m/s
1
2941.2
2
3816.8
3
3225.8
1
2597.4
2
4065.0
3
4000.0
As the fluid pressure in the well decreases, the
radial stresses rr at points M and N, equal to the
pressure in the well, decrease also, and the circumferential stresses r/ increase, as they are
8.1 Results of Physical Modeling of Resistance …
proportional to the difference between the value
of rock pressure and the value of fluid pressure in
the well.
As the measurement of p-waves velocity
demonstrates that rocks of all five lithotypes are
elastically isotropic, the stresses in the
bottom-hole zone of the uncased borehole,
assuming that the initial stress field is a state of
uniform compression, are determined by the
solution of the Lamé problem (3.12).
Horizontal and vertical stresses in real massifs
may be not equal. In that case rock fracture in the
vicinity of the well will depend on its values and
orientation of the principle stresses acting in the
massif with respect the well axis. However, the
aim of the experiments was to demonstrate the
influence of strength anisotropy on the stability
and fracture of rocks near the well. Therefore, in
order to identify this fracture mechanism, the
rock specimens were loaded in experiments
according to the program based on the assumption of isotropy of the initial stress field. The
magnitude of the drawdown Dpw ¼ p0 pw in
the well is related to the circumferential effective
stress s/ acting on its wall, as follows
Dpw ¼ p0 þ q s/ =2
ð8:1Þ
where p0 is reservoir pressure
In the course of the experiments, one part of
the tested specimens was loaded in accordance
with the conditions corresponded to point M
(top), and the other part was loaded in accordance with the conditions corresponded to point
N (side), Fig. 3.8. This was achieved by
Fig. 8.9 The pictures of
tested specimens:
F2-PsG-5-1, F2-PsG-5-2
115
corresponding placing the specimen in the loading unit of the TILTS. When modeling stress
state corresponding to point N, the specimen was
positioned in such a way that the 2-axis of the
installation, along which the load grew monotonously during the experiment (Fig. 5.8), coincided with the core axis. When modeling stress
state corresponding to point M, the specimen was
positioned so that the 2-axis of the unit was
perpendicular to the core axis.
In the course of each experiment, the strains in
each of the three directions together with the
changes in the permeability of the specimen on
one of its axes were recorded over time. Permeability of the specimens corresponding to points
N and M was measured along different axes of
the specimen: along the core axis for point M,
and normally to the core axis for point N, i.e. in
both cases permeability was measured in the
direction toward the well.
Modeling of the process of pressure decrease
at the bottom-hole of a horizontal well on TILTS
has revealed a number of interesting facts. First
of all, the maximum compressive stress s2 , at
which the specimens were destroyed, depended
significantly on the location of the point on the
horizontal well contour. Failure of the specimens
located at the upper point M on the well contour
occurred at much lower values s2 than the
specimens located at the side point N.
As an example below are given the results of
testing of two specimens of clay sandstone
made cut from the same piece of core:
F2-PsG-5-1 and F2-PsG-5-2 (Fig. 8.9). Table 8.2
116
8
Results of Tests of Rock Specimens by Using TILTS
Table 8.2 P-wave velocities in specimens of the Filanovsky field
№
specimen
Litotype
Specimen
axis
Velocity,
m/s
F2-PsG-5-1
Clay
sandstone
1
2273
2
2564
3
2439
1
2273
2
2326
3
2439
F2-PsG-5-2
Clay pot
shows the results of measurement of p-wave
velocities for each of the three axes of the specimens (the 1-axis of the core). It is seen that the
velocities for both specimens are close and do not
depend much on the direction of measurement.
This suggests that the elastic properties of the
specimens are identical and isotropic.
Specimen F2-PsG-5-1 was tested under conditions corresponding to the top point of the well
contour (point M), and specimen F2-PsG-5-2
was tested under conditions corresponding to the
side point N.
Figures 8.10 and 8.11 show the deformation
curves obtained during testing of the specimens.
Curves e1 ; e2 ; e3 correspond to the strains along
axes 1,2,3, the loading parameter, depicted along
the axis of ordinates, corresponds to monotonically increasing stress s2 .
It is followed from Figs. 8.10, 8.11 and formula (8.1) that specimen F2-PsG-5-1, modeling
the upper position on the well contour (point M),
failed at the load corresponding to the drawdown
on the bottom-hole of 2.4 MPa, while specimen
F2-PsG-5-2, corresponding to the lateral position
on the well contour (point N) failed at a much
higher drawdown, 6.1 MPa.
A similar situation was observed when testing
specimens of other lithotypes: specimens tested
under the conditions of the upper point of the
well contour were destroyed much earlier than
specimens tested under the conditions of the
lateral point of the contour. Table 8.3 presents
the test results of the specimens with indication
of their location on the contour of a horizontal
well: the stress value s2 at which the specimens
Fig. 8.10 Deformation curves of specimen F2-PSG-5-1
Fig. 8.11 Deformation curves of specimen F2-PSG-5-2
were failed, and the values of drawdown Dpw at
the bottom-hole that correspond to this stress.
Thus, testing of specimens of all lithotypes
revealed a significant anisotropy of their strength
properties, which strongly affects the fracture of
rocks in the vicinity of a horizontal well. Specimens located at the top of the horizontal well
contour were destroyed much earlier than those
at the side of the contour. This fact is the most
amazing because the measurements of p-wave
velocities did not reveal anisotropy of their
elastic properties.
As it is shown above (see Chap. 3), the fracture on the horizontal well contour in the presence of anisotropy of strength is most likely to
start at the points on the well contour, located at
8.1 Results of Physical Modeling of Resistance …
117
Table 8.3 Results of tests of the Filanovsky field specimens
r2 s2 MPa
Dpw
MPa
№ specimen
Litotype
Depth selection, m
Location
F2-P4 (c)
Sandstone
1358.2
Top
39
3.5
F4- P -9-2 (b)
Sandstone
1412.7
Side
63
15
F2- PcA-1 (c)
Alevritis sandstone
1352.1
Top
36
1.5
F4- PcA-7-2 (b)
Aleuritic sandstone
1405.7
Side
66
15
F 2-PcG-5-1 (c)
Clay bandstone
1367.3
Top
36
2.5
F 2-PcG-5-2 (b)
Clay sandstone
1367.3
Side
46
6.2
F 8-Pp-2 (c)
Layering
1444.8
Top
38
1.8
angles of 30°–45° to the vertical axis of the well
contour, Fig. 3.8. Therefore, for more accurate
estimation of the allowable drawdown on the
bottom-hole of a horizontal well, which would
not causes damage to the borehole walls, it is
necessary to carry out a physical modeling of the
process of deformation and fracture of rocks at
various points of the well contour.
Experiments on rock specimens from the
reservoir showed also significant anisotropy of
filtration properties, despite the isotropy of elastic
properties and lack of visible layering. In the
horizontal plane, the permeability of rocks was
much higher than in the vertical direction.
Modeling of the process of pressure decrease
at the bottom-hole of a horizontal well by the
TILTS has shown that non-even stress state in
the vicinity of the well occurring due to drawdown can cause a significant change in permeability in this zone—both to its decrease and
increase. The increase in permeability, sometimes very significant, was observed mainly
while specimens were tested in accordance with
the program corresponding to their location at the
top of the horizontal well contour.
In Fig. 8.12 the change in permeability of
F2-PsG-5-1 and F2-PsG-5-2 specimens during
the tests is shown. The permeability of the
specimen F2-PsG-5-1 before fracture increased
dramatically, while the permeability of the
specimen F2-PsG-5-2 in the process of loading
first decreased, then increased, but in the end
practically did not change.
In our view, this fact should be taken into
account and requires further study. Such studies
may reveal the stress states that need to be created in the borehole zone to increase permeability
and well productivity. This issue is particularly
relevant for the operation of horizontal wells.
The carried out researches allow drawing an
important conclusion related to the choice of
deformation, strength and filtration characteristics of rocks of productive layers to be of the
priority subject for experimental determination
for creation and filling the geomechanical model
of a field. The current traditional set of such data
is based on the assumption that the elastic and
strength properties of rocks are isotropic
(Young’s modulus, Poisson ratio, constants of
Mohr-Coulomb
failure
criterion
or
Drucker-Prager criterion, etc.).
For determining these traditional parameters,
Karman’s type installations, which do not allow
creating the true stress states occurring in the
reservoirs in the vicinity of wells, mainly used.
However, the deformation, strength and filtration
properties of rocks significantly depend on the
level and type of stresses created in them.
Therefore, it can be stated that conclusions and
recommendations on ensuring rock stability
within the bottom-hole formation zone, maximum allowable drawdown and flow rates, which
are obtained on the basis of geomechanical
models that do not account for the anisotropy of
the deformation and strength properties of rocks,
and the dependence of their filtration properties
on the stress-strain state may be quite far from
reality and do not solve the main problem—to
reduce risks and improve efficiency during the
operations of wells.
118
8
Results of Tests of Rock Specimens by Using TILTS
Fig. 8.12 Changes in
permeability of specimens
during the experiment
8.2
Determination of Parameters
of Models of Plastic
Deformation for Transverse
Isotropic Reservoir and Host
Rocks
Parameters of plastic models (generalized model
of Hill’s plasticity in forms of Lui-Huang-Staut
(LHS) and Caddel-Raghava-Atkins (CRA), and
combined criterion based on two fracture mechanisms)—the parameters of the criteria for
elastic-inelastic transition and plasticity potentials—were determined by analyzing the results
of tests of rock specimens, conducted bu TILTS
for productive and host formations of a number
of oil and gas fields that have shown an anisotropy of elastic and strength properties:
–
–
–
–
–
–
–
Vostochno-Surgutskoye
Konitlorskoye
Russkinskoye
Fedorovskoye
Talakanskoye
Filanovsky
Kainsaiskoye.
For determining the parameters of plasticity
models for each lithotype, if possible, information
from all the available tests was used. The data were
based mainly on the results of two most frequently
used loading programs: triaxial experiments and
physical modeling of deformation processes in the
vicinity of wells (generalized shear). It should be
noted that the type and parameters of the loading
programs used for each set of tests was often
determined according to current tasks and optimization of core material consumption, so the full
set of tests was conducted not for all types of rocks.
Also, it was not always possible to determine all
set of parameters for some rocks.
The results of determining parameters of
plasticity models for each field are described
below.
The results are presented in the form of tables
(Tables 8.4, 8.5, 8.6, 8.7, 8.8, 8.9 and 8.10) and
diagrams of dependence of the critical stress on
the angle between the maximal compressive
stress and the layering plane (Figs. 8.13, 8.15,
8.17, 8.19, 8.22, 8.25 and 8.28), diagrams of the
dependence of shear stress intensity corresponding to elastic-inelastic transition on the first
invariant of stress tensor (sum of compressive
stresses in three axes) (Fig. 8.14, 8.16, 8.18,
8.20, 8.23, 8.26 and 8.29), and diagrams of the
dependence of the critical stress on the value of
lateral compression in the experiments on triaxial
compression (Fig. 8.21, 8.24, 8.27 and 8.30).
In each of Tables 8.4, 8.5, 8.6, 8.7, 8.8, 8.9
and 8.10, the sequence number of the specimen
is indicated in Column 1. Column 2 depicts the
angle in degrees between axis of the maximum
compressive stress and the plane of isotropy
(layering plane). Columns 3–4 indicate the values of the principle stresses (with a reverse sign)
corresponding to inelastic-inelastic transition in
the coordinate system associated with the axes of
the specimen. Columns 6–9 depict components
8.2 Determination of Parameters of Models of Plastic Deformation …
119
Table 8.4 Critical stresses in specimens of the Vostochno-Surgutskoye field cut out at different angles to bedding and
parameters of plasticity models
1
2
3
№
Angle
Stresses in the
coordinate system of
the specimen
Stresses in the coordinate system
associated with the axes of
symmetry of the rock
Relative error
s1
s2
s3
s′11
s′13
DP
Comb
%
%
Degree
4
5
6
7
8
s′22
9
s′33
10
MPa
MPa
MPa
MPa
MPa
MPa
MPa
0
72
36.5
1
72
36.5
1
0
2
0
71
36
1
71
36
1
0
3
15
70
35.5
1
65.4
35.5
5.6
17.25
4
30
63
35
1
47.5
35
16.5
26.8
5
30
59
35
1
44.5
35
15.5
25.1
6
45
59
35
1
30
35
30
7
45
62
35
1
31.5
35
31.5
8
60
58
35
1
15.25
35
43.75
24.7
9
75
72
36.5
1
5.76
36.5
67.2
17.75
10
90
72
36.5
1
1
36.5
72
0
1
11
12
13
LHS
CRA
%
%
−0.55
0.985
−0.14
−1.99
−1.86
−0.49
19.8
18.3
−6.18
12.8
7.73
9.85
−17.1
−21.1
29
−1
−4.6
−10
30.5
−9.57
19.4
14.1
9.26
−24.6
−27.7
23.4
25.2
0.205
0.205
0.205
Sum of squared deviations
0.39
3.14
−0.55
0.985
24
25.5
Table 8.5 Critical stresses in specimens of the Konitlorskoye field cut out at different angles to bedding and
parameters of plasticity models
1
2
3
№
Angle
Stresses in the
coordinate system of
the specimen
4
5
s1
s2
Degree
MPa
MPa
6
7
8
9
10
11
12
13
Comb
LHS
CRA
%
%
%
Stresses in the coordinate system
associated with the axes of
symmetry of the rock
Relative error
s3
s′11
s′22
s′33
s′13
DP
MPa
MPa
MPa
MPa
MPa
%
1
0
85
43
1
85
43
1
0
21.0
36.6
2
0
67
34
1
67
34
1
0
−5.0
−17.1
3
30
59
32
5
45.5
32
18.5
23.4
7.05
−14.7
−35.8
4
30
62
32
2
47
32
17
25.98
−5.71
−2.6
−12.5
5
30
65
33
1
49
33
17
27.71
−11.6
4.8
2.8
6
30
57
32
7
44.5
32
19.5
21.65
17.3
−22.9
−50.1
7
45
70
35.5
1
35.5
35.5
35.5
34.5
−9.9
10.9
17.7
8
45
82
41.5
1
41.5
41.5
41.5
40.5
−17.8
30.4
72.4
9
60
90
67.75
1
23.25
67.75
67.75
38.58
10
90
100
50.5
1
1
50.5
100
0
Sum of squared deviations
13.0
0
0
10.0
23.3
57.7
−7.7
−32.1
37.7
153
120
8
Results of Tests of Rock Specimens by Using TILTS
Table 8.6 Critical stresses in specimens of the Russkinskoye field cut out at different angles to bedding and
parameters of plasticity models
1
2
3
№
Angle
Stresses in the
coordinate system of
the specimen
Stresses in the coordinate system
associated with the axes of
symmetry of the rock
Relative error
s1
s′11
Degree
4
s2
5
6
s3
7
s′22
8
9
10
11
12
13
s′33
s′13
DP
Comb
LHS
CRA
%
%
%
MPa
MPa
MPa
MPa
MPa
MPa
MPa
%
1
0
119
60
1
119
60
1
0
0
2
30
88
44.5
1
66.25
44.5
22.75
37.67
−11
3
45
67
37
7
37
37
37
30
14
9.7
−1.5
−3.4
11.9
5.4
−15.6
9.7
4
45
67
37
7
37
37
37
30
14
−15.6
5
30
80
40.5
1
20.75
40.5
60.25
34.2
3.4
1.6
5.3
6.3
Sum of squared deviations
0
0.01
25.1
Table 8.7 Critical stresses in specimens of the Fedorovskoye field cut out at different angles to bedding and
parameters of plasticity models
1
2
3
№
Angle
Stresses in the
coordinate system of
the specimen
Stresses in the coordinate
system associated with the axes
of symmetry of the rock
Relative error
s1
s2
s3
s′11
s′22
s′33
s′13
DP
MPa
MPa
MPa
MPa
MPa
MPa
MPa
%
Degree
1
0
2
30
4
5
6
7
8
9
10
11
12
13
Comb
LHS
CRA
%
%
%
92.1
46.55
1
92.1
46.5
1
0
80
40.5
1
60.2
40.5
20.75
34.2
−0.39
3.77
−35
9.33
13
3
45
74
37.5
1
37.5
37.5
37.5
36.5
−2.06
4.01
0.9
4
45
71.2
36.1
1
36.1
36.1
36.1
35.1
1.45
−0.21
16
5
90
113
57
1
1
57
113
0
−1.1
6.6
42
6
0
2
2
49
2
2
49
0
−0.7
2.63
−88
7
0
10
10
77
10
10
77
0
5.3
1.98
0.4
8
0
20
20
114
20
20
114
0
9.3
6.07
104
2.09
243
Sum of squared deviations
1.18
of the stress tensor (with the opposite sign) in the
coordinates associated with the principle axes of
elasticity tensor of the specimen (s012 ¼ s023 ¼ 0
for all specimens) obtained by the standard procedure of tensor rotating
r011 ¼ r1 cos2 u þ r3 sin2 u
r022 ¼ r2
r033 ¼ r1 sin2 u þ r3 cos2 u
r013 ¼ ðr1 r3 Þ sin u cos u
ð8:2Þ
0.07
Columns (10–13) indicate the calculated values
to be zeroed according to the criteria used (more
precisely, minimized, accounting for the
approximate and empirical nature of the criteria,
as well as the unavoidable measurement error)
for the parameters obtained by the least square
method:
Column 10 contains the ratios of the left-hand
side of expression for the Drucker-Prager criterion (1.26) and the absolute value of the mean
principle stress
8.2 Determination of Parameters of Models of Plastic Deformation …
121
Table 8.8 Critical stresses in specimens of the Talakanskoye field cut out at different angles to bedding and parameters
of plasticity models
1
2
3
№
Angle
Stresses in the
coordinate system of
the specimen
Stresses in the coordinate system
associated with the axes of
symmetry of the rock
Relative error
s1
s2
DP
DP
DP
DP
s′13
DP
Comb
MPa
MPa
MPa
MPa
MPa
MPa
MPa
%
%
63.7
32.35
1
63.7
32.35
1
0
Degree
4
5
6
7
8
9
10
11
12
13
LHS
CRA
%
%
−1.99
−41.7
1
0
2
30
57
29
1
43
29
15
24.25
0.62
2.53
2.26
3
30
57.7
29.35
1
43.53
29.35
15.18
24.55
−0.50
3.83
5.91
4
45
53.6
27.3
1
27.3
27.3
27.3
26.3
0.25
−5.16
−7.08
5
90
105
53
1
1
53
105
0
1.8
0.45
14.59
6
0
2
2
62.5
2
2
62.5
0
−18.7
−4.04
−31.93
7
0
10
10
85
10
10
85
0
14.5
6.15
16.00
8
0
20
20
100
20
20
100
0
−8.06
−2.67
13.01
Sum of squared deviations
6.31
.007
1.13
34.9
Table 8.9 Critical stresses in specimens of the V. Filanovsky field cut out at different angles to bedding and
parameters of plasticity models
1
2
3
№
Angle
Stresses in the
coordinate system of
the specimen
4
5
6
s1
s2
s3
s′11
s′22
s′33
s′13
DP
Comb
LHS
CRA
Degree
MPa
MPa
MPa
MPa
MPa
MPa
MPa
%
%
%
%
38
20
38
20
2
0
−0.0
−0.61
2
7
8
9
Stresses in the coordinate
system associated with the axes
of symmetry of the rock
10
11
12
13
Relative error
1
0
2
90
64
33
2
2
33
64
0
−0.0
3
0
27
2
2
27
2
2
0
4.3
6.6
4
0
51
10
10
51
10
10
0
5
0
75
20
20
75
20
20
0
Sum of squared deviations
1.22
−3.3
−4.1
0
0.9
−7.6
0
0.3
1.1
122
8
Results of Tests of Rock Specimens by Using TILTS
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
2
2
2
pffiffiffi ðr1 r3 Þ þ ðr1 r2 Þ þ ðr2 r3 Þ þ Bðr1 þ r2 þ r3 Þ A =jr2 j
6
ð8:3Þ
Table 8.10 Critical stresses in specimens of the Kainsaiskoye field cut out at different angles to bedding and
parameters of plasticity models
1
2
3
№
Angle
Stresses in the
coordinate system of
the specimen
Stresses in the coordinate
system associated with the
axes of symmetry of the rock
Relative error
s1
s3
s′11
s′22
s′33
s′13
DP
MPa
MPa
MPa
MPa
MPa
%
156
0
−0.6
Degree
4
5
s2
MPa
MPa
2
2
6
2
9
10
11
12
13
Comb
LHS
CRA
%
%
%
−3.1
3.2
0
2
0
5
5
165
5
5
165
0
−0.06
−1.9
2.2
3
0
15
15
220
15
15
220
0
5.8
−10.1
−12.2
4
0
18
18
191
18
18
191
0
−0.9
9.0
4.6
5
0
83
2
2
83
2
2
0
15.6
15.6
6
0
126
10
10
126
10
10
0
−10.4
−10.4
7
0
164
20
20
164
20
20
0
−18.3
−18.3
8
0
2
92
184
2
92
184
0
40
36
9
0
2
129
129
2
129
129
0
52
49
10
90
2
75
148
148
75
2
0
1.8
−0.6
24
27
Fig. 8.13 Dependence of
critical stress s1 on angle u
for rocks of the
Vostochno-Surgutskoye field
2
8
1
Sum of squared deviations
156
7
−10.6
0.68
8.2 Determination of Parameters of Models of Plastic Deformation …
Fig. 8.14 Dependence of
critical intensity ri on the first
invariant s0 ¼ s1 þ s2 þ s3 for
rocks of the
Vostochno-Surgutskoye field
Fig. 8.15 Dependence of
critical stress s1 on the angle
u for rocks of the
Konitlorskoye field
Fig. 8.16 Dependence of the
critical intensity of shear
stress ri on the first stress
invariant of stress tensor s0 for
rocks of the Konitlorskoye
field
123
124
8
Results of Tests of Rock Specimens by Using TILTS
Fig. 8.17 Dependence of the
value of critical stress s1 on
the angle u between the
direction s1 and the formation
plane for the rocks of the
Russkinskoye field
Fig. 8.18 Dependence of
critical intensity of shear
stresses si on the first
invariant of stress tensor s1 for
rocks of the Russkinskoye
field
To determine parameters of Drucker-Prager
criterion only the tests, for which fracture in
planes of weakness was considered to be switched off, were chosen, namely tests conducted
using programs of three-axis compression and of
generalized shear for the maximum compressive
stresses acing normally to layering.
Column 11 contains the ratios of the left-hand
side of expression for the fracture criterion along
planes of weakening (1.37) and value of stresses
normal to the planes of weakening
s sc
tgqc
rn
ð8:4Þ
Here only the tests, for which fracture in
planes of weakness was supposed to be dominant, were chosen, i.e. tests for which the angle
between the principle compressive stresses and
the layering plane differed from 0° and 90°.
Column 12 contains the left-hand side of
Lui-Huang-Staut criterion (LHS) (1.45)
8.2 Determination of Parameters of Models of Plastic Deformation …
FL ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
GLð13Þ ðr22 r33 Þ2 þ GLð13Þ ðr11 r33 Þ2 þ GLð12Þ ðr11 r22 Þ2 þ 2LLð13Þ r231
þ BLð1Þ ðr11 þ r22 Þ þ BLð3Þ r33 1
Fig. 8.19 Dependence of
critical stress s1 on angle u
for rocks of the Fedorovskoye
field
Fig. 8.20 Dependence of
critical intensity of shear
stress si on the first invariant
of stress tensor s0 for rocks of
the Fedorovskoye field
125
ð8:5Þ
126
Fig. 8.21 Dependence of
critical stress s3 on lateral
compression s1 ¼ s2 for the
second type of loading
program for rocks of
the Fedorovskoye field
Fig. 8.22 Dependence
critical stress s1 on angle u
for rocks of the Talakanskoe
field
Fig. 8.23 Dependence of
critical intensity of shear
stresses si on the first
invariant of stress tensor s0
for rocks of the Talakanskoe
field
8
Results of Tests of Rock Specimens by Using TILTS
8.2 Determination of Parameters of Models of Plastic Deformation …
Fig. 8.24 Dependence of
critical stress s3 on lateral
compression s1 ¼ s2 for rooks
of the Talakanskoe field
Fig. 8.25 Dependence of
critical stress s1 on the angle
u for rocks of the Filanovsky
field
127
128
8
Results of Tests of Rock Specimens by Using TILTS
Fig. 8.26 Dependence of
critical intensity of shear
stress si on the first invariant
of stress tensor s0 for rocks of
the Filanovsky field
Fig. 8.27 Dependence of
critical stress s3 on lateral
compression s1 ¼ s2 for rocks
of the Filanovsky field
Column 13 contains the left-hand side of
Caddel-Raghava-Atkins (CRA) criterion (1.51)
F C ¼ GCð13Þ ðr22 r33 Þ2 þ GCð13Þ ðr11 r33 Þ2
þ GCð12Þ ðr11 r22 Þ2 þ 2LCð13Þ r231
þ BCð1Þ ðr11 þ r22 Þ þ BCð3Þ r33 1 ¼ 0
ð8:6Þ
8.2 Determination of Parameters of Models of Plastic Deformation …
129
Fig. 8.28 Dependence of
critical stress s1 on the angle
u for rocks of the
Kainsaiskoye field
layering plane. The points correspond to the
experimental data.
Solid lines correspond to the combined criterion: horizontal sections correspond to failure
due to maximum stress, according to the loading
program
r3 ¼ r03 ;
r2 ¼ ðr1 þ r3 Þ=2
ð8:7Þ
(for the majority of cases r03 ¼ 1 MPa) and criterion (1.26) the critical stress is
r1 ¼
Fig. 8.29 Dependence of critical stress s3 on lateral
compression s1 ¼ s2 for rocks of the Kainsaiskoye field
For the last two cases, the results of all
available experiments were used.
Figures 8.13, 8.15, 8.17, 8.19, 8.22, 8.25 and
8.28 depict the dependences of critical stress r1
on angle u between the direction of r1 and the
2A þ ð1 þ 3BÞr03
1 3B
ð8:8Þ
According to the loading program, fracture
along planes of weakening is possible both after
and before the beginning of rising of the intermediate principle stress. In the first case, the principle
stresses are determined by (8.7); in the second
case, the principle stresses are determined as
follows
130
8
r2 ¼ r02 ;
r1 ¼ 2r02 r3
Results of Tests of Rock Specimens by Using TILTS
ð8:9Þ
Values of r02 differ for various lithotype and
correspond to in situ stresses. Together with the
criterion (1.34) relations (8.7), (8.9) allow to
obtain the value of critical stress r1 . For the first
and second cases we have, respectively
r1 ¼
sc þ r03 ðsin u cos u þ tgqc cos2 uÞ
ð8:10Þ
sin u cos u tgqc sin2 u
r1 ¼ r02 þ
sc þ r02 tg qc
sin 2u þ tgqc cos 2u
ð8:11Þ
Thus, the lines corresponding to the combined
criteria generally consist of three sections:
– horizontal lines, for angles close to 0° and 90°;
– adjacent lines, for which the critical stress is
determined by formula (8.10);
– the central regions, for which the critical
stress is determined by formula (8.11).
Dotted lines correspond to the critical stress
according to LHS criterion (1.45). These lines
generally consist of two sections corresponding
to different stages of the loading program: the
stage of increasing intermediate principle stress
(8.7) and the stage of its constancy (8.9),
respectively
pffiffiffi
ð a þ 2cÞr03 þ 2
pffiffiffi
r1 ¼
ð8:12Þ
a 2b
1 þ r02 2BLð1Þ þ BLð3Þ
ð8:13Þ
r1 ¼ r02 þ pffiffiffi a þ BLð3Þ BLð1Þ cos 2u
Fig. 8.30 Dependence of critical stress s1 on lateral
compression s3 ¼ s2 for rocks of the Kainsaiskoye field
a ¼ 5GLð13Þ ; þ GLð12Þ cos 2u þ 2LLð13Þ sin 2u
1
L
2
þ cos u þ BLð3Þ sin2 u
b ¼ Bð1Þ
2
1
L
2
þ sin u þ BLð3Þ cos2 u
c ¼ Bð1Þ
2
ð8:14Þ
The dashed lines correspond to the critical
stresses according to CRA criterion (1.51). The
lines also generally consist of two sections corresponding to two described above stages of
loading program, respectively
r1 ¼
2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
aðb þ cÞr03 þ a þ b2 þ ar03 þ 2b
a
ð8:15Þ
where
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
iffi
r1 ¼
r02
þ
BCð3Þ BCð1Þ
2
cos2 2u þ 4a 1 þ r02 2BCð1Þ þ BCð3Þ
2a
BCð3Þ þ BCð1Þ
ð8:16Þ
8.2 Determination of Parameters of Models of Plastic Deformation …
where
a ¼ 5GCð13Þ ; þ GCð12Þ cos 2u þ 2LCð13Þ sin 2u
1
C
2
þ cos u þ BCð3Þ sin2 u
b ¼ Bð1Þ
2
1
C
2
þ sin u þ BCð3Þ cos2 u
c ¼ Bð1Þ
2
ð8:17Þ
For the rocks of a number of fields (Fedorovskoye, Talakanskoye, the Kainsayskoye) the
dependences of the critical stress r3 on lateral
compression r1 ¼ r2 (or for critical stress r1 on
lateral compression r2 ¼ r3 for the Filanovsky
field and the Kainsayskoye field) are presented
for tests according to programs of the second type.
The values according to the criteria used in the
first case are the following: for Drucker-Prager
criterion:
pffiffiffi
A þ 2B þ 1 3 jr1 j
pffiffiffi
jr3 j ¼
1 3B
ð8:18Þ
for LHS criterion:
1 þ 2BLð1Þ þ BLð3Þ jr1 j
qffiffiffiffiffiffiffiffiffiffiffiffiffi
jr3 j ¼ jr1 j þ
2GLð13Þ BLð3Þ
ð8:19Þ
for CRA criterion:
jr 3 j ¼ jr 1 j þ
BCð3Þ þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
i
C
C
C
BC2
ð3Þ þ 8Gð13Þ 1 þ 2Bð1Þ þ Bð3Þ jr1 j
4GCð13Þ
ð8:20Þ
The values according to the criteria used in the
first case are the following: for Drucker-Prager
criterion:
pffiffiffi
A þ 2B þ 1 3 jr3 j
pffiffiffi
jr1 j ¼
1 3B
for LHS criterion:
ð8:21Þ
1 þ 2BLð1Þ þ BLð3Þ jr3 j
jr1 j ¼ jr3 j þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
GLð13Þ þ GLð12Þ BLð1Þ
131
ð8:22Þ
for CRA criterion:
jr1 j ¼ jr3 j
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
iffi
C
C
C
C
BCð1Þ þ BC2
þ
4
G
þ
G
þ
B
1
þ
2B
ð1Þ
ð13Þ
ð12Þ
ð1Þ
ð3Þ jr3 j
þ
2 GCð13Þ þ GCð12Þ
ð8:23Þ
Results of Determination of Strength Properties of the Vostochno-Surgutskoe Field
The tests were carried out according to the program of loading of the type of generalized shear
on the specimens made cut at angles of 0°, 15°,
30°, 45°, 60°, 75°, 90° to layering (Karev et al.
2016b). Effective values were r02 ¼ 35 MPa,
r03 ¼ 1 MPa. A total of 10 specimens were tested.
The measured values of critical stresses depending on the angle together with the results of calculations are presented in Table 8.4 and
Fig. 8.13. Only the tests, for which fracture in
planes of weakness was not essential was used for
calculating constants of Drucker-Prager criterion
(column 10). Such tests include specimens 9–10,
for which the maximum compressive stress was
applied normally to the layering (specimen 10) or
close to the normal (specimen 9), and specimens
1–3, for which the maximum compressive stress
was applied along the layering (or at the angle of
15°, specimen 3). Specimens 1–3 were included,
because the critical stresses for them practically
coincide with the critical stresses for specimens
9–10 that were subjected to the maximum compression normally to layering. The minimum
standard deviation corresponds to A ¼ 8:05 MPa
and B ¼ 0:25. It should be noted that since for all
five used results the loading program essentially
coincided, and the values of critical stresses were
very close, the obtained values for two parameters
are not very reliable—close values to minimize
the standard deviation of the value (8.3) may be
obtained with other combinations of parameters.
132
For calculation parameters of criterion of
fracture along planes of weakening (Column 11),
the results of tests were chosen, for which the
effect of weakening was predominant (specimens
4–8). The minimum standard deviation corresponds to sc ¼ 21:9 MPa, tgqc ¼ 0:05; the latter
value was chosen because the angle of friction
cannot be negative (negative values gave less
standard deviation), so a small positive value was
chosen.
All 10 test results were used for determining
parameters of LHS and CRA criteria (columns
12, 13). The minimum standard deviation corresponds to the following values of the
parameters
8
Results of Tests of Rock Specimens by Using TILTS
It is seen from Fig. 8.13 that for the rocks of
the Vostochno-Surgutskoye field the combined
criterion appears to give the best correlation with
the experimental results. The results obtained
using LHS and CRA criteria are close to each
other, but the curves they define are too smooth
and do not accurately describe the transition to
inelasticity for angles u close to 0° and 90°. In
addition, according to these criteria, the points of
change of mechanisms s02 ¼ const to s03 ¼ const
are shifted outwards comparing to the experimental points.
Figure 8.14 depicts the dependence of the
critical intensity of shear stresses si on the first
invariant of stress tensor s0 ¼ s1 þ s2 þ s3 . It can
be seen from the plot that it is impossible to
GLð13Þ ¼ 0:00053 MPa2 ; GLð12Þ ¼ 0:00053 MPa2 ; describe the entire set of experiments by using
Drucker-Prager criterion: the points correspondLLð13Þ ¼ 0:0032 MPa2 ; BLð1Þ ¼ 0:0092 MPa1 ;
ing to the loads applied inclined to layering lie
BLð3Þ ¼ 0:0092 MPa1
well below the line plotted according to this
criterion.
for LHS criterion and
Konitlorskoye Field
C
2
C
2
Gð13Þ ¼ 0:000135 MPa ; Gð12Þ ¼ 0:000135 MPa ; The tests were carried out according to the loading
program of the generalized shear on 10 specimens
LCð13Þ ¼ 0:00079 MPa2 ; BCð1Þ ¼ 0:0001 MPa1 ;
made cut at angles of 0° (2 specimens), 30° (4
BCð3Þ ¼ 0:00901 MPa1
specimens), 45° (2 specimens), 60°, 90° (one
specimen each) to layering. Values of applied
for CRA criterion. Here, by virtue of the stresses were s02 ¼ 32, s03 ¼ 1 MPa (Karev et al.
observed symmetry, it was set
2016b). Dependence of the measured and calculated values of the critical stresses on the angle
GLð13Þ ¼ GLð12Þ ; BLð1Þ ¼ BLð2Þ and
are presented in Table 8.5 and Fig. 8.15.
GLð13Þ ¼ GLð12Þ ; BLð1Þ ¼ BLð2Þ :
Only one test result (Specimen 10) was
selected for calculation parameters of DruckerFigure 8.13 depicts the dependence of critical Prager criterion (Column 10), for which the effect
stress s1 on angle u between the direction of s1 of weakening along layering on fracture can be
and the plane of layering. From Fig. 8.13 it is considered negligible. Since it is not possible, in
seen that dependence of critical stress on the principle, to determine two parameters from a
angle is close to symmetric with respect to angle single measurement, therefore one of them
of 45°. For the conditions of the Vostochno- ðB ¼ 0:25Þ was chosen by analogy with the
Surgutskoye field, the critical stresses, according properties of similar lithotypes. The minimum
to the combined criterion, contain all three types standard deviation (its equality to zero) correof segments, although the segment corresponding sponded to A ¼ 11;625 MPa.
To calculate parameters of the criterion for
to case (8.10) are very small (corresponding to
angles of about 20° and 68°). The critical stress weakening planes (Column 11), the results of
according to LHS and CRA criteria also contain tests were chosen for which the influence of
weakening along the layering planes on fracture
both constancy of s02 and constancy of s03 .
8.2 Determination of Parameters of Models of Plastic Deformation …
was predominant (specimens 3–9). The minimum root-mean-square deviation corresponds to
sc ¼ 18:4 MPa and tgqc ¼ 0:356 MPa
All 10 test results were used for determining
parameters of LHS and CRA criteria (columns
12, 13). The minimum standard deviation corresponds to the following values of the
parameters
GLð13Þ ¼ 0:000141 MPa2 ;
GLð12Þ ¼ 0:000289 MPa2 ;
LLð13Þ ¼ 0:0008 MPa2 ; BLð1Þ
BLð3Þ ¼ 0:00594 MPa1
¼ 0:0084 MPa1 ;
for LHS criterion, and
GCð13Þ ¼ 0:000149 MPa2 ;
LCð13Þ ¼ 0:0007 MPa2 ;
GCð12Þ ¼ 0:00007 MPa2 ;
BCð1Þ ¼ 0:000429 MPa1 ;
BCð3Þ ¼ 0:0129 MPa1
for CRA criterion. Here, no symmetry with
respect to the angle of 45° was observed, so all
constants were assumed to be different.
Figure 8.15 depicts dependence of the critical
stress s1 on the angle u between the direction s1
and the layering plane. Figure 8.15 demonstrates
that for the conditions of the Konitlorskoye field
the critical stresses according to the combined
criterion contain all three types of segments, and
the segments corresponding to case (8.10) are
sufficiently extended. The curves corresponding to
the critical stresses according to the LHS and CRA
criteria also contain both segment of constancy s02
and constancy of s03 . The figure demonstrates that it
is difficult to select the criterion for the conditions
of this field, which gives the best match with the
experimental results. The combined criterion gives
better results for angles greater than 30°, but
overestimates the critical stress for small angles.
The results obtained by LHS and CRA criteria are
close to each other, give the correct qualitative
description, the correct assessment of the position
of the point of switch the mechanisms s02 ¼ const
and s03 ¼ const, however, do not accurately
describe the critical stresses for angles u close
to 90°.
133
Figure 8.16 depicts dependence of the critical
intensity of shear stresses ri on the first invariant
of stress tensor s0 . Points corresponding to the
critical stresses due to mechanism of fracture
along planes of weakening are shown blank.
It is seen from the plot that it is impossible to
describe the entire set of experiments by using
Drucker-Prager criterion—the points corresponding to the load applied inclined to layering
lie below the line corresponding to this criterion.
It is also seen that the choice of the parameter
value B ¼ 0:25 in the Drucker-Prager criterion is
qualitatively correct: larger values of the parameter would lead to a stronger inclination of the
dependence, and the points corresponding to the
criterion of fracture along the planes of weakening would fall on the line, or would be higher,
which contradicts to the nature of the observed
phenomenon.
Russkinskoye Field
The tests were carried out according to the program of loading of type of generalized shear on 5
specimens made cut under angles of 0°, 30°, 45°
(2 specimens), 60° to layering (Karev et al.
2016). The applied values of stresses were
s02 ¼ 37 MPa, s03 ¼ 1 MPa. Measured and calculated values of the critical stresses as functions of
the angle are presented in Table 8.6 and
Fig. 8.17.
Only one test result (Specimen 1) was suitable
for determining parameters of Drucker-Prager
criterion (Column 10), for which weakening
along layering on fracture did not affect the
critical stress. Since it is not possible to determine, in principle, two parameters from a single
test, one of them ðB ¼ 0:25Þ was chosen by
analogy with the properties of rocks of similar
lithotypes. The minimum standard deviation (its
equality to zero) corresponded to A ¼ 14 MPa.
To calculate parameters of the criterion for
weakening planes (Column 11), the results of the
tests were chosen for which the effect of weakening along the planes of layering on fracture was
decisive (specimens 2–5). The minimum standard
deviation corresponds to sc ¼ 32:4 MPa, tgqc ¼
0:05, the latter being chosen as a small positive
134
8
value according to a consideration that the friction
angle cannot be negative (negative values gave
less standard deviation).
The results of all 5 tests were used for determining parameters of LHS and CRA criteria
(columns 12, 13). The minimum standards
deviations correspond to the following values of
parameters
It can be seen from the plot that it is impossible to describe the entire set of experimental
data using Drucker-Prager criterion only—the
points corresponding to the load inclined to
layering lie well below the curve built according
to this criterion. It is also seen from the plot that
the choice of parameter B ¼ 0:25 in the
Drucker-Prager criterion is qualitatively correct:
larger values of the parameter would lead to
stronger inclination of the line, and the points
corresponding to the criterion of fracture along
the planes of weakening would fall on the line,
which contradicts the nature of the phenomenon.
GLð12Þ ¼ GLð13Þ ¼ 4:64 105 MPa2 ;
LLð13Þ ¼ 3:95 104 MPa2 ;
BLð1Þ ¼ BLð3Þ ¼ 0 MPa1
for LHS criterion and
GCð13Þ ¼ GCð12Þ ¼ 4:86 105 MPa2 ;
LCð13Þ ¼ 0:0004 MPa2 ;
BCð1Þ ¼ BCð3Þ ¼ 2:77 104 MPa1
for CRA criterion. Here, because of the small
amount of the experimental data and the
observed symmetry, it was set
GLð13Þ ¼ GLð12Þ ; BLð1Þ ¼ BLð2Þ and
GLð13Þ ¼ GLð12Þ ; BLð1Þ ¼ BLð2Þ :
Figure 8.17 depicts the dependence of the
critical intensity of shear stress s1 on angle u
between the direction s1 and the layering plane.
It is seen from Fig. 8.17 that for the conditions of the Russkinskoye field the critical
stresses according to the combined criterion
contain all three types of sections, the sections
corresponding to case (8.10) being sufficiently
wide. The results obtained using LHS and CRA
criteria are close to each other. Due to the small
amount of data, it is difficult to select a criterion
that matches the experimental results the best.
Figure 8.18 depict the dependence of the
critical intensity of shear stresses ri on the first
invariant of stress tensor s1 . Points corresponding
to the critical stresses due to mechanism of
fracture along planes of weakening are shown
blank.
Results of Tests of Rock Specimens by Using TILTS
Fedorovskoye Field
The tests were conducted according to two types
of loading programs. Loading program of the
type of generalized shear was used for testing 5
specimens made cut at angles of 0°, 30°, 45° (2
specimens), 90° to layering (Kovalenko et al.
2016; Karev et al. 2016, 2017a). The applied
values of stresses were s02 ¼ 35 MPa, s03 ¼ 1
MPa. Both experimental and calculated values of
the critical stresses for angles as functions of the
angle are presented in Table 8.7 and Fig. 8.19.
Besides, program of triaxial loading was used
for one specimen, for which three critical points
corresponding to lateral stresses of 2, 10, 20 MPa
were obtained. The results are also presented in
Table 8.7.
Results of experiments according to the program on triaxial loading (three points) and on
generalized shear (Specimen 5), in which the
maximum compressive stress was applied normally to layering, were used for determining
parameters of Drucker-Prager criterion (Column
10), The minimum standard deviation corresponded to A ¼ 13:9 MPa, B ¼ 0:25:
To calculate parameters of the criterion for
weakening planes (Column 11), the results of the
tests were chosen for which the effect of the
weakening along the planes of layering on fracture was predominant (specimens 2–4). The
minimum standard deviation corresponds to sc ¼
32 Mpa and tgqc ¼ 0:1.
8.2 Determination of Parameters of Models of Plastic Deformation …
Results of all 8 tests were used for determining parameters of LHS and CRA criteria (columns 12, 13). The minimum standard deviation
corresponded to the following values of the
parameters
GLð13Þ ¼ 0:00108 MPa2 ;
GLð12Þ ¼ 0:00168 MPa2 ;
LLð13Þ ¼ 0:00425 MPa2 ;
BLð1Þ ¼ 0:02 MPa1 ;
BLð3Þ ¼ 0:022 MPa1
135
data and values calculated according to the used
criteria. Figures 8.20 and 8.21 demonstrate that
for the conditions of the field the most adequate
description is provided by LHS criterion.
Accounting for the influence of compression
using the CRA criterion appeared somewhat
difficult.
The combined criterion yields good results for
angles far from 0° and 90°, but does not allow
describing the observed difference in critical
stresses for angles close to 0° and 90°.
for LHS criterion and
GCð13Þ ¼ 0:000332 MPa2 ;
LCð13Þ ¼ 0:0014 MPa2 ;
GCð12Þ ¼ 0:00013 MPa2 ;
BCð1Þ ¼ 0:022 MPa1 ;
BCð3Þ ¼ 0:026 MPa1
for CRA criterion. Here, no symmetry with
respect to the angle of 45° was observed, so all
constants were assumed to be different.
Figure 8.19 depicts the dependence of the
critical stress s1 on angle u between the direction
s1 and the layering plane. Figure 8.19 demonstrates that for the conditions of the
Fedorovskoye field, the critical stresses according
to the combined criterion contain only two types
of sections: sections corresponding to s02 ¼ const
(8.9) are absent. The lines of critical stresses
according to LHS and CRA criteria also do not
contain sections corresponding to this mode.
Figure 8.20 shows the dependence of the
critical intensity of shear stress si on the first
invariant of stress tensor s0 . The points corresponding to loading program of generalized
shear type are shown blank. It can be seen from
the plot that it is impossible to describe the entire
set of experiments using Drucker-Prager criterion
only—the points corresponding to loading
inclined to layering lie below the curve corresponding to this criterion. However, the rightmost point corresponding to the maximum
compressive stresses applied normally to layering lies on the line.
Figure 8.21 shows the dependence of critical
axial stress r3 on lateral compression s1 ¼ s2 for
the second type of load test for both experimental
Talakanskoye Field
The tests were conducted on two types of loading
programs. According to the loading program of
the type of generalized shear 5 specimens
made cut at angles of 0°, 30° (2 specimens), 45°,
90° to layering were tested (Karev et al. 2016).
Values of applied stresses were s02 ¼ 20 MPa,
s03 ¼ 1 MPa. The measured values of the critical
stresses depending on the angle and the results of
calculations are presented in Table 8.8 and in
Fig. 8.22. One specimen was tested according to
the program of triaxial loading with lateral
compression, for which 3 points of transition to
inelasticity, corresponding to lateral stresses 2,
10, 20 MPa, were obtained. These results are
also presented in Table 8.8.
Results of experiments according to the program of triaxial loading (three points) and results
of experiment on generalized shear (Specimen
5), in which the maximum compressive stress
was applied normally to layering, were used for
determining parameters of Drucker-Prager criterion (Column 10), The minimum standard deviation corresponded to the following values:
A ¼ 24; B ¼ 0:17.
To calculate parameters of the criterion for
weakening planes (Column 11), the results of the
tests were chosen for which the effect of the
weakening along the planes of layering on fracture was predominant (specimens 2–4).
The minimum standard deviation corresponds
to sc ¼ 22 MPa, tgqc ¼ 0:16.
All 8 test results were used to determine
parameters of LHS and CRA criteria (columns
136
8
12, 13). The minimum standard deviation corresponds to the following values of the parameters
Figures 8.23 and 8.24 demonstrate that for
conditions of the Talakanskoe field the most
adequate description is provided by LHS criterion.
CRA criterion overestimates the critical stresses
for small angles between maximum compressive
stress and layering. The combined criterion gives
good results for angles far from 0° and 90°, but
does not allow describing the observed difference
in critical stresses for angles close to 0° and 90°.
GLð13Þ ¼ 0:00034 MPa2 ;
GLð12Þ ¼ 0:00012 MPa2 ;
LLð13Þ ¼ 0:00144 MPa2 ;
BLð1Þ ¼ 0:0036 MPa1 ;
BLð3Þ ¼ 0:0097 MPa1
for LHS criterion and
GCð13Þ ¼ 0:00026 MPa2 ;
GCð12Þ ¼ 0:00001 MPa2 ;
LCð13Þ ¼ 0:00134 MPa2 ;
BCð1Þ ¼ 0:0073 MPa1 ;
BCð3Þ ¼ 0:019 MPa1
for CRA criterion. Here, no symmetry with
respect to the angle of 45° was observed, so all
constants were assumed to be different.
Figure 8.22 shows the dependence of critical
stress s1 on angle u between the direction of s1
and the layering plane. For the conditions of the
Talakanskoe field, the critical stresses according
to the combined criteria contain only two types
of sections: sections corresponding to s02 ¼ const
(8.9) are absent. The lines corresponding to
critical stresses according to LHS and CRA criteria also do not contain sections corresponding
to this mode.
Figure 8.23 shows the dependence of intensity of critical shear stress si on the first invariant
of stress tensor s0 . The points corresponding to
the loading by generalized shear program are
shown blank. It can be seen from the plot that it
is impossible to describe the entire set of experiments using Drucker-Prager criterion—the
points corresponding to the load inclined to
layering lie well below the curve corresponding
to this criterion. However, the rightmost point,
corresponding to the maximum compressive
stresses applied normally to layering, lies close to
the line.
Figure 8.24 depicts dependence of critical
stress s3 on the lateral compression s1 ¼ s2 for
the second type of loading program. Experimental data and calculated values according to
the criteria used are presented: DP, (8.18); LHS,
(8.19); CRA, (8.20).
Results of Tests of Rock Specimens by Using TILTS
Filanovsky Field
The tests were conducted according to two types
of loading programs. Two specimens made cut at
angles of 0°, 90° to the layering were tested
according to loading program of the type of
generalized shear (Karev et al. 2016a, 2017a,
2018a, c). The applied values of stresses were
s02 ¼ 20 MPa, s03 ¼ 2 MPa. Measured values of
critical stresses and the results of calculations are
presented in Table 8.9 and Fig. 8.25. Triaxial
tests, for which 3 points of transition to inelasticity corresponding to lateral stresses 2, 10,
20 MPa, were also carried out. The maximum
compressive stress was applied along an axis
parallel to the layering. The results are presented
in Table 8.9.
For calculating parameters of Drucker-Prager
criterion (Column 10), only one result for test on
Specimen 2 was selected, for which the maximum compressive stress was applied normally to
layering. In other tests, weakening planes could
play a significant role in reduction of the critical
stresses. Since two parameters cannot be found
from a single relation, parameter B was estimated
on the base of the following considerations. For
all experiments, except specimen 2, the weakening planes participated in the process of transition to inelasticity, therefore, on their
hypothetical switching off, the corresponding
critical stresses would be greater. Therefore, in
Fig. 8.25 the line, corresponding to DP criterion,
passing through the point corresponding to
specimen 2 should pass above all other points,
which corresponds to 0:055\B\0:23. By
analogy with the rocks of similar lithotypes,
value B ¼ 0:15 was accepted. The corresponding
8.2 Determination of Parameters of Models of Plastic Deformation …
value A ¼ 16:15 MPa is obtained from the condition of the line passing through the experimental point.
Parameters of criterion of fracture along
planes of weakening (column 11) were not
determined due to the lack of experimental data.
All 5 test results were used for determining
parameters of LHS and CRA criteria (columns
12, 13). In this case, better fit was obtained if the
target function was chosen as the sum of the
ratios of squares of deviations of the maximum
stress and the maximum stress, rather than the
sum of the squares of deviations from the criteria
in the forms of (8.5) or (8.6). The minimum
standard deviation corresponds to the following
values of the parameters
137
criterion only: the points, for which the transition
to inelasticity is conditioned by weakening along
layering lie below the line obtained according to
this criterion. Figure 8.27 shows the dependence
of critical stress s1 on lateral compression s3 ¼ s2
for the second type of loading program. Experimental data and calculated values according to
the criteria used are presented: DP, (8.18), LHS,
(8.19), CRA, (8.20).
It is followed from Figs. 8.25, 8.26 and 8.27
that for conditions of the Filanovsky field transition to inelastic state is described equally adequate by both LHS and CRA criteria. The latter
gives a slightly lower estimate for high compression values (right points in Fig. 8.27).
Kainsaiskoye Field
GLð12Þ ¼ 0:0013 MPa2 ; Tests were carried out in accordance with three
types of loading programs (Karev et al. 2018d).
LLð13Þ ¼ 0:016 MPa2 ; BLð1Þ ¼ 0:019 MPa1 ;
According to the loading program of the type of
BLð3Þ ¼ 0:033 MPa1
generalized shear two specimens made cut at the
angles of 0°, 90° to the layering were tested. The
applied values of stresses were s02 ¼ 20 MPa,
for LHS criterion and
s03 ¼ 2 MPa. The measured values of the critical
C
2
C
2
Gð13Þ ¼ 0:005 MPa ; Gð12Þ ¼ 0:0046 MPa ; stresses and the results of calculations are presented in Table 8.10 and Fig. 8.28. Two speciLCð13Þ ¼ 0:033 MPa2 ; BCð1Þ ¼ 0:17 MPa1 ;
mens were also tested in accordance with the
BCð3Þ ¼ 0:37 MPa1
triaxial loading program with constant lateral
compression, for each of which three critical
for CRA criterion. Here, parameters LLð13Þ and points were obtained. For one of the specimens
the maximum stress was applied parallel to layLCð13Þ could not be determined from the available
ering, lateral stresses being 2, 10, 15 MPa. For
experiments, so their values were chosen in such the second specimen, the maximum stress was
a way that the dependence of critical stresses s1 applied normally to layering, lateral stresses
on angle u between the direction s1 and the being 2, 10, 20 MPa. One more specimen was
layering plane (Fig. 8.25) were similar to the tested according to the same loading program
dependencies typical to rocks of similar with lateral compression of 18 MPa and maxilithotypes.
mum stress applied parallel to the layering. The
Figure 8.26 shows the dependence of critical program js j ¼ js j [ js j was also used. The
3
2
1
intensity of shear stresses si on the first invariant results are presented in Table 8.10.
of stress tensor s0 . The point corresponding to the
For calculating parameters of Drucker-Prager
load according to program of generalized shear criterion (Column 10), the results of the tests of
js1 j [ js2 j [ js3 j is shown as blank. It can be the specimens were chosen for which the effect
seen from the plot that it is impossible to describe of the weakening of the layering planes on
the whole set of experiments by Drucker-Prager fracture can be assumed to be inessential. These
GLð13Þ ¼ 0:0027 MPa2 ;
138
Table 8.11 Constants
used in calculation for the
Kirinskoye field
8
E,
MPa
m
6000
0.3
Field
Results of Tests of Rock Specimens by Using TILTS
Cycle 1
Cycle 3
si ,
MPa
sm ,
MPa
si ,
MPa
sm ,
MPa
36
14
77
46
ss
a
Ep
18
1.28
1370
Combined criterion
A
B
sc
tgqc
MPa
–
MPa
–
Vostotchno–Surgutskoye
8.05
0.25
21.9
0.05
Konitlorskoe
11.6
0.25
18.4
0.356
Russkinskoye
14
0.25
32.4
0.05
Fedorovskoe
13.9
0.25
32
0.1
Talakanskoye
24
0.17
22
0.16
Filanovsky
16.2
0.15
–
–
Kainsaiskoye
62
0.175
–
–
Field
Modified Hill’s criteria
LHS criterion
G13
G12
CRA criterion
L13
104 MPa−2
B1
B3
G13
G12
104 MPa−1
104 MPa−2
L13
B1
B3
104 MPa−1
Vostotchno–Surgutskoye
5.3
5.3
32
92
92
1.4
1.4
7.9
1.0
1.0
Konitlorskoe
1.41
2.89
8
8.4
59.4
1.49
0.7
7
4.29
129
Russkinskoye
0.46
0.46
3.95
0
0
0.48
0.48
4
2.77
2.77
Fedorovskoe
10.8
16.8
42
200
220
3.3
1.3
14
200
260
Talakanskoye
3.4
1.2
14.4
36
97
2.6
0.1
13.4
73
190
Filanovsky
27
13
160
190
330
50
46
330
1700
3700
Kainsaiskoye
0.83
1.1
4.9
31
60
3.4
3.3
21
450
1000
include tests 1–4, for which the maximum compressive stress was applied normally to layering,
and the results of testing of specimen 8. The
minimum standard deviation corresponds to
A ¼ 62 MPa, B ¼ 0:175.
Calculations of parameters of the criterion for
weakening planes (column 11) were not carried
out due to the lack of experimental data.
The results of all 10 tests were used for
determining parameters of LHS and CRA criteria
(columns 12, 13). In this case, as in case of the
Filanovsky field, better fit was obtained if the
target function was chosen as the sum of the
ratios of squares of deviations of the maximum
stress and the maximum stress, rather than the
sum of the squares of deviations from the criteria
in the forms (8.5) or (8.6). The minimum standard deviations correspond to
GLð13Þ ¼ 8:3 105 MPa2 ;
GLð12Þ ¼ 1:1 104 MPa2 ;
LLð13Þ ¼ 4:9 104 MPa2 ;
BLð1Þ ¼ 0:0031 MPa1 ;
BLð3Þ
¼ 0:0060 MPa
1
References
139
for LHS criterion and
GCð13Þ ¼ 0:00034 MPa2 ;
LCð13Þ ¼ 0:0021 MPa2 ;
GCð12Þ ¼ 0:00033 MPa2 ;
BCð1Þ ¼ 0:045 MPa1 ;
BCð3Þ ¼ 0:1 MPa1
for CRA criterion. Here, parameters LLð13Þ and
stress levels. Constants for the investigated rock
of Kirinskoye gas condensate field (sandstone)
are presented in Table 8.11.
Summary of Field Data
These results of determination of parameters of
plasticity criteria for each field are presented in
the form of summarizing tables.
LCð13Þ could not be found from the available
experiments, so their values were chosen in such
a way that the dependence of critical stresses s1
on angle u between the direction s1 and the
layering plane (Fig. 8.28) were similar to the
dependencies typical for rocks of similar
lithotypes.
Figures 8.29 and 8.30 depict the dependencies
of critical stress s3 on lateral compression
s1 ¼ s2 , and s1 on s3 ¼ s2 , respectively for triaxial tests. Experimental data and calculated
values according to the used criteria are presented: DP, (8.18), LHS, (8.19), CRA, (8.20). it
is seen from Figs. 8.28, 8.29 and 8.30 that for the
conditions of the Kainsaiskoye field the transition to inelastic state is described equally adequate by criteria LHS and CRA. Both criteria
overestimate the critical stress for the program of
generalized shear when the maximum compressive stress applied along layering. This, however,
may be due to the lower initial strength or
damage of the specimen.
Kirinskoye Field
The rocks of the Kirinskoye field (sandstone) did
not exhibit any anisotropic properties of permeability, strength or elasticity. Therefore, the
Drucker-Prager criterion was used for their
modeling (Karev et al. 2015, 2018a; Zhuravlev
et al. 2012). Three specimens were selected to
determine the properties, two of which were
tested under the generalized shear type program
with simultaneous measurement of permeability,
and the third specimen was tested according triaxial test in three cycles corresponding to different lateral compression. The rocks of two
different types manifested different dependence
of permeability on the stress state, but the transition to inelastic state took place at the same
References
Karev VI, Kovalenko YuF (2006) Dependence of the
bottom-hole formation zone permeability on the pressure drawdown and bottom-hole design for different
types of rocks. Tekhnologii TEK (Technologies of the
Fuel and Energy Complex) 6:59–63 (in Russian)
Karev VI, Kovalenko YuF (2013) Triaxial loading system
as a tool for solving geotechnical problems of oil and
gas production. In: True triaxial testing of rocks. CRC
Press, Balkema, Leiden: 301–310
Karev VI, Kovalenko YuF, Zhuravlev AB, Ustinov KB
(2015) Filtering model in a well taking into account
permeability dependence on the stresses. Processy v
Geosredach (Processes in Geomedia) 4(4):35–44 (in
Russian)
Karev VI, Kovalenko YuF, Sidorin YV, Ustinov KB
(2016a) Geomechanical modeling of processes in
bottom-hole zone. Monitoring. Nauka i Tekhnologii
(Monitor Sci Technol) 3(28):85–91 (in Russian)
Karev VI, Klimov DM, Kovalenko YuF, Ustinov KB
(2016b) Fracture of sedimentary rocks under complex
triaxial stress state. Mech Solids 51(5):522–526
Karev VI, Kovalenko YuF, Ustinov KB (2017a) Modeling deformation and failure of anisotropic rocks
nearby a horizontal well. J Mining Sci 53(3):425–432
Karev VI, Kovalenko YuF, Sidorin YuV, Stepanova EV,
Ustinov KB (2017b) Modeling of fluid seapage in well
at great depths accounting the anisotropy of reservoir
strength properties. Processy v Geosredakh (Processes
in Geomedia) 2(11):512–521 (in Russian)
Karev VI, Klimov DM, Kovalenko YuF, Ustinov KB
(2018a) Modelling of mechanical and filtration processes near the well with regard to anisotropy. J Phys
Conf Ser 991(1):012039. https://doi.org/10.1088/
1742-6596/991/1/012039
Karev VI, Klimov DM, Kovalenko YuF (2018b) Modeling geomechanical processes in oil and gas reservoirs
at the true triaxial loading apparatus. In: Physical and
mathematical modeling of earth and environment
processes. Springer geology, vol 30, pp 336–349.
https://doi.org/10.1007/978-3-319-77788-7_35
Karev VI, Klimov DM, Kovalenko YuF, Ustinov KB
(2018c) Modeling of deformation and filtration processes near wells with emphasis of their coupling and
effects caused by anisotropy. In: Physical and
140
8
Results of Tests of Rock Specimens by Using TILTS
mathematical modeling of earth and environment
processes. Springer geology, vol 30, pp 350–360.
https://doi.org/10.1007/978-3-319-77788-7_36
Karev VI, Klimov DM, Kovalenko YuF, Ustinov KB
(2018d) Modeling of deformation and filtration processes near producing wells: influence of stress state
and anisotropy. In: Litvinenko V (ed) Proceedings of
symposium on geomechanics and geodynamics of
rock masses (EUROCK2018), Taylor & Francis
Group, London 2:1381–1386
Karev VI, Klimov DM, Kovalenko YuF, Ustinov KB
(2018e) Physical modeling of real geomechanical
processes by true triaxial apparatus. In: Litvinenko V (ed) Proceedings of symposium on geomechanics and geodynamics of rock masses
(EUROCK2018), Taylor & Francis Group, London
2:1375–1380
Karev VI, Klimov DM, Kovalenko YF, Ustinov KB
(2018f) Fracture model of anisotropic rocks under
complex loading. Phys Mesomech 21(3):216–222
Klimov DM, Karev VI, Kovalenko YuF, Ustinov KB
(2015) Interaction of stress-strain state and filtration in
rocks. In collection: Actual problems in mechanics:
50 years of Ishlinsky Institute for Problems in
Mechanics RAS. M. Science: 489–508. ISBN
978-5-02-039181-9 (in Russian)
Kovalenko YuF, Ustinov KB, Zhuravlev AB (2016)
Stress-strain state in the vicinity of perforated well
taking into account inelastic deformation. Processy v
Geosredakh (Processes in Geomedia) 1(5):69–76 (in
Russian)
Zhuravlev AB, Karev VI, Kovalenko YuF, Sidorin YuV,
Sirotin AA, Ustinov KB (2012) On plastic deformation of rocks. Determination of plastic characteristics
according to triaxial tests. In: Collection of papers of
the 3d International Conference on Topical Problems
of Continuum Mechanics. October 2012 (Tsakhkadzor, Armenia): 238–242. ISBN 978-9939-63-129-5
(in Russian)
9
Mathematical Modeling
of Mechanical and Filtration
Processes in Near-Wellbore Zone
Two groups of problems related to mathematical
modeling are considered. The first group is
devoted to finding the stress–strain state near the
wellbores both for the purposes of determination
of technological parameters that ensure the
wellbores stability and for initiation the process
of controlled fracture (method of directed
unloading of the formation, see Chap. 10) (Karev
et al. 2017; Klimov et al. 2009). The second
group of problems is related to calculation well
production rates with accounting for stress state
influence on filtration processes (Zhuravlev et al.
2014; Karev et al. 2015, 2017; Klimov et al.
2015). Both groups of problems were solved for
the same set of geometries. Problems of determining the stress state were solved both for the
absence of filtration (in order to identify the
conditions necessary to create a permeable zone),
and in its presence. In the latter version, the
mathematical statement became identical to the
mathematical statement of the second type
problems.
The following bottom-hole configurations
were studied:
1) an uncased wellbore (well radius R ¼ 0:1 m)
(Zhuravlev et al. 2014);
2) a wellbore with perforation slot (Fig. 9.1);
3) a wellbore with perforation holes (Fig. 9.2).
Geometric parameters of the model of wellbore
with perforation slot are the following: external
dimensions in plane, a ¼ 10 m; wellbore radius,
R = 0.1 m; perforation slot depth, L = 0.46 m;
© Springer Nature Switzerland AG 2020
V. Karev et al., Geomechanics of Oil and Gas Wells,
Advances in Oil and Gas Exploration & Production,
https://doi.org/10.1007/978-3-030-26608-0_9
perforation slot thickness, h = 0.02 m; depth of
the design zone in the direction of the well axis,
hz ¼ 0:05 m.
Geometric parameters of the model with perforation holes are the same for a simple borehole
(the parameters L, h are not used): external
dimensions in plane, a = 10 m; the borehole
radius, R = 0.1 m; the length of the perforation
hole, L = 0.2 m; the diameter of the perforation
hole at the borehole contour, d1 = 0.04 m; the
diameter of the perforation hole at the end of the
borehole, d2 = 0.03 m; external dimensions in
the direction of the borehole, hz = 25 m.
9.1
Calculation of the Inelastic
Deformation Zone
in the Absence of Filtration
In calculations the schemes based on
elastic-plastic deformation models that implement either isotropic Drucker-Prager law or the
modified anisotropic Hill plastic flow theory with
plastic potential (1.76) were used and additional
condition (1.77) and purely isotropic hardening
ðaij ¼ 0Þ. Transition to the plastic state was
determined by the plasticity condition (1.48).
For the conditions of the Fedorovskoye field
(Kovalenko et al. 2016), the problems of determining the stress-strain state in the vicinity of a
well with one or two perforation holes according
to the Drucker-Prager inelastic deformation
141
9 Mathematical Modeling of Mechanical and Filtration Processes …
142
Fig. 9.1 A wellbore with a
perforation slot
Fig. 9.2 Wells with
perforation holes
R
L
h
d1
d2
9.1 Calculation of the Inelastic Deformation Zone in the Absence of Filtration
Table 9.1 Model parameters used in calculations for
rocks of the Fedorovskoye field
E (MPa)
ss (MPa)
a
Ep (MPa)
12,600
25
1.25
1300
sn ¼ rn þ p
ð9:1Þ
where p is the fluid pressure; rn is normal total
stress, sn ; q\0, P > 0.
Calculations were carried out for the following conditions: normal effective stress on the
walls of the uncased wellbore and the surface of
the perforation holes sn ¼ 0; radial effective
stress sr ðRk Þ ¼ q þ pc (pc is pressure in the
wellbore) on the external boundary; radius of the
external boundary Rk ¼ 10Rc (Rc being radius of
Fig. 9.3 Finite element mesh
the well). In practice, the concept of drawdown
pressure Dpc in a well is often used instead of
well pressure pc
DPc ¼ P0 Pc
model were solved. For comparison, the problems were also solved as purely elastic. The
elastic, plastic and strength constants of the rock
calculated using the results of the three-axis test
are given in Table 9.1. In the calculation it was
assumed that the rock mass in its initial state is
subjected to uniform equi-component compression of the rock pressure q ¼ ch, where c is the
specific gravity of the overlying rocks, h is the
collector depth.
In a plane with normal n, the rock skeleton is
subjected to the action of effective stress
143
ð9:2Þ
where p0 is the reservoir pressure. The other
parameters used in calculations were: depth H =
2750 m; rock pressure q = 63 MPa (at density c
= 2.3 g/cm3); reservoir pressure P0 = 27.5 MPa.
Figure 9.3 depicts the mesh for FEM to solve
the problem of stress distribution in the vicinity
of the well with two perforation holes in the form
of cones. Figures 9.4 and 9.5 depict the distributions of intensity of shear stresses in the
vicinity of the wellbore with one and two perforation holes for elastic and elastic-plastic
solutions.
The value of pressure drawdown in the well
was assumed to be Dpc = 10 and 25 MPa. Figure 9.6 shows the distribution of plastic deformations in the vicinity of a well with two
perforation holes for two values of pressure
drawdown: 10 and 25 MPa. Calculations of
stress states occurring in the vicinity of a perforated wellbore performed using elastic
andelastic-plastic models (Figs. 9.4 and 9.5),
suggests the following:
Accounting for inelastic behavior of the rock
leads to expanding the zones of stress
144
9 Mathematical Modeling of Mechanical and Filtration Processes …
Fig. 9.4 Stress distribution
around the well with one
perforation hole; elastic
solution (a); elastic-plastic
solution (b)
Fig. 9.5 Stress
distribution in the vicinity
of a well with two
perforation holes; elastic
solution (a); elastic-plastic
solution (b)
concentration in the vicinity of the wellbore with
both one and two perforation holes compared to
the elastic solution.
The presence of plastic deformation zones in
the vicinity perforation holes significantly reduces the level of stresses.
With the growth of the pressure drawdown,
the zone of plastic deformations occurring in the
vicinity
of
perforation
holes
expands
significantly.
On the basis of the obtained results it is possible to draw a number of practical conclusions,
which should be considered during development
of methods of increasing oil and gas recovery
and maintenance of accident-free methods of
drilling and operation of oil and gas wells.
One of the main parameters determining the
wellbore output when using any method of its
operation is the allowable level of pressure
drawdown. It is clear that the greater the
depression (i.e. the lower the pressure in the well)
the greater the output to be achieved. However
the risks associated with the failure of the wellbore walls and sand production are rising, as the
stresses near the wellbore increasing with the
drawdown increase. This issue has become
especially urgent recently, when almost all fields
are developed with application of drilling of
horizontal wells, the productive sections if which
remain open (not cemented). In this case, the
pressure drop in the well is directly transferred to
the surrounding rock and causes an increase in
stresses.
The above results demonstrate that the modeling of the stress-strain state in the vicinity of
the well in the elastic formulation significantly
overestimate the value of stresses occurring in
this zone, thereby overestimating the risk of
9.1 Calculation of the Inelastic Deformation Zone in the Absence of Filtration
145
Fig. 9.6 Distribution of
plastic deformations;
depression Dpc = 10 MPa (a);
depression Dpc = 25 MPa (b)
walls destruction and underestimating the value
of the ultimate pressure drawdown and, as a
consequence, the maximum allowable output of
the well.
Thus, in order to issue reliable forecast recommendations for accident-free drilling and well
operation, as well as to achieve maximum well
production rates, it is necessary to conduct a set
of experimental and theoretical studies. They
should include testing of core material from the
fields under study with the special equipment for
determining the elastic-plastic and strength
parameters of reservoir rocks under conditions of
true three-axis loading, as well as modeling the
stress-strain state in the vicinity of wells
accounting for inelastic behavior of rocks.
Calculations were also carried out for open
hole and perforated hole configurations. The data
corresponded to the rocks of the Fedorovskoye
field (the properties of rocks are given above),
but under other conditions of occurrence, which
is reflected in the change of boundary conditions:
the normal stress at the outer boundary 50 MPa.
Isolines of intensity of pressure and intensity of
plastic strains for the considered configurations
are presented on Fig. 9.7.
The presence of a perforation cut leads to an
increase in stress concentration. For an uncased
well, which is a cylindrical hole, the influence of
the anisotropy of plastic properties is expressed
as the deviation of the isolines of intensities
stresses and strain from the concentric circles.
The deviations of the isolines of stress intensity
and plastic strains are directed in opposite
directions from the concentric circles corresponding to the isotropic case.
Figure 9.8 depicts the boundaries of the zones
within which the criteria (1.26), (1.38) and (1.76)
are satisfied, computed within the framework of
the elasticity for the uncased well. The configurations of these zones, computed according to
both criteria, as well as by using the finite element method, are very similar.
9.2
Calculation of Zone of Inelastic
Deformation in Case
of Filtration; The Algorithm
For each configuration, modeling is carried out
by several steps (Ustinov 2016; Karev et al.
2018a, b).
The first stage consists in solving the filtration
problem in order to determine the first iteration of
the field of fluid pressure. The second stage
consists in solving the problem of poroplasticity
divided into three substages: (i) solving the
problem of uncoupled poroelasticity for the
146
9 Mathematical Modeling of Mechanical and Filtration Processes …
Fig. 9.7 Isolines of intensity
of stresses for open borehole
(a), intensity of plastic strains
for open borehole (b),
intensity of stresses for
borehole for wellbore with a
perforation cut (c) intensity of
plastic strains for borehole for
wellbore with a perforation
cut (d)
Fig. 9.8 Boundaries of the zones of fulfillment of the
combined criterion (1.26), (1.38) (dotted line) and the
modified Hill criterion (1.76) (dashed) calculated using
elastic model. Solid line corresponds to the well contour
calculated pore pressure field; (ii) calculating the
plastic properties of the media as a function of
coordinates on the base of the calculated pore
pressure and stress fields; (iii) solving the problem of poroplasticity for the calculated plastic
properties. The third stage consists in calculating
permeability as a function of coordinates by
applying the experimentally obtained law of
change of permeability on the stress intensity.
Then the difference between the solution of the
filtration problem and the solution obtained at the
previous stage was calculated; if the difference
(as a parameter determining the difference, the
total inflow into the well was used) exceeded the
specified value ðe [ 1%Þ, the stress state is
recalculated for the distribution of the fluid
pressure obtained during the previous iteration.
At the final stage the problem of filtration and
determination of the flow rate is solved.
Block-scheme of the algorithm is given on
Fig. 9.9. the simulation were carried out in 3-D
using meshes, corresponding to one quarter of
the domain in question for the described configurations of bottom-hole. In case of perforation
cuts the total number of nodes and elements was
44,001 and 22,356, respectively.
For Kirinskoye field (Karev et al. 2018d;
Zhuravlev et al. 2012), two different types of
rocks from two reservoir layers were selected for
modeling. The first core sample had been taken
from depth h = 2776 m. Rock pressure at this
depth is q ¼ ch = −63.8 MPa, where c = 2.3 103 kg/m3 is the average specific gravity of
overlying rocks, fluid pressure is p0 = 27.7 MPa.
The rock skeleton at this depth is subject to
effective stresses
sn ¼ q þ aP p0 ¼ 36:1 MPa;
sn \0;
p[0
ð9:3Þ
where sn is the normal stress acting on the plane
with normal n. The second core sample had been
taken from depth 2862 m. The rock skeleton at
this depth is subjected to stress of 37.2 MPa. In
accordance with that values the boundary conditions for the normal stress and pore pressure on
the external boundaries was applied. The stress
9.2 Calculation of Zone of Inelastic Deformation in Case …
147
Fig. 9.9 Block-scheme of
the algorithm
and pressure on the well contour were assumed
to be zero.
For the rocks of the first type (sample 1) the
permeability decreased monotonously with the
increase of shear stresses up to the value
scr ¼ 63 MPa, after which, it dropped almost to
zero. For the second type of rocks (sample 2), the
permeability decreased monotonously with the
increase of shear stresses up to the value
scr ¼ 55:4 MPa, after which it began to increase
sharply. Experimental dependencies, together
with approximating lines obtained with the least
squares method are shown at Fig. 9.10.
1 0:0079s 7 105 s2 s\63 MPa
0:05
s 63 MPa
1 0:0062s
s\55:4 MPa
¼
0:0526s 2:26 s 55:4 MPa
k1 =k10 ¼
k2 =k20
ð9:4Þ
The presented plots, in particular, demonstrate
that the change in permeability correlates with
the stress state rather than with inelastic strains
only. In any case, the drop in permeability begins
before noticeable inelastic deformations appear.
Therefore, the dependence on stress has been
chosen as the determining factor for permeability
change. For other rocks, the proper choice may
be different.
Modelling was also carried out for the conditions of the V. Filanovsky and Kainsaiskoye
fields (Karev et al. 2016, 2018c, e). The programs of loading and changes in permeability
due to stress change are shown in Figs. 9.11 and
9.12.
For V. Filanovsky field the radial stress and
pore pressure at the external boundary were 31
and 13 MPa, respectively; the radial stress and
pore pressure at the wellbore wall is 0 MPa. For
148
9 Mathematical Modeling of Mechanical and Filtration Processes …
Fig. 9.10 Dependencies of
permeability on shear stress in
experiments simulating the
stress state in the vicinity of
the wellbore for rock of the
first (a) and second (b) types
of the Kirinskoye field
Kainsaiskoye field the radial stress and pore
pressure at the external boundary were 141 and
61 MPa, respectively. On the surfaces normal to
the well axis, the absence of normal to the
boundaries displacements and the conditions of
non-permeability were set.
Calculation Results
The values of outputs normalized to the output
into the ideal well (the permeability of the formation in the vicinity of the well for which is
supposed to be constant, homogeneous and equal
to the natural permeability) without a perforation,
9.2 Calculation of Zone of Inelastic Deformation in Case …
149
Fig. 9.11 Loading program (a) and change in permeability (b); numerical modeling of the stress state in the
bottom-hole zone for the rock of the V. Filanovsky field
Fig. 9.12 Loading program (a) and change in permeability (b); numerical modeling of the stress state in the
bottom-hole zone for the rock of the Kainsaiskoye field
modeled accounting various factors are presented
in Table 9.2.
The distribution of stress intensity, plastic
strain intensity and pore pressure for some
characteristic combinations of the used models
are presented in Fig. 9.13 and Figs. 9.14, 9.15,
9.16, 9.17 for V. Filanovsky and Kainsaiskoye
fields, respectively (Karev et al. 2018d).
During calculations, it was found that for used
parameter values, the account for elastic
anisotropy did not cause changes in the stress
distribution comparing to the isotropic model;
the account for plastic anisotropy caused changes
in the distribution of inelastic deformations
(fracture zones), but also did not cause noticeable
change in the calculated flow rate. Accounting
for filtration anisotropy has led to significant
change in the output and a change in the pore
pressure distribution.
9 Mathematical Modeling of Mechanical and Filtration Processes …
150
Table 9.2 Outputs normalized to the output into the ideal wellbore
Geometry
Kirinskoe
(1-st)
Deformation
Model Filtration Model
Elasticity
Plasticity
No dependency
Isotropic
permeability
Open
borehole
Isotropic
No
1
0.72
Perforation slot
Isotropic
No
1.5
0.89
Open borehole
Isotropic
No
1
0.94
Perforation slot
Isotropic
No
1.5
1.43
Isotropic (DP)
0.86
Isotropic (DP)
Kirinskoye (2-st)
1.11
Isotropic (DP)
0.92
Isotropic (DP)
V. Filanovsky
1.3
Open borehole
Isotropic
No
Perforation slot
Isotropic
No
Two perforation hole
Isotropic
No
Open borehole
Isotropic
Isotropic
1
0.77
1.27
0.87
Anisotropic
0.79
Anisotropic
0.89
1.04
0.81
1
1.07
1.08
Anisotropic
1.07
1.08
Anisotropic
Isotropic
1.07
1.08
Cut along layering
Anisotropic
Anisotropic
Cut normal to layering
Anisotropic
Anisotropic
Anisotropic
Kainsainskoye
Anisotropic
permeability
0.83
Anisotropic
Fig. 9.13 Stress intensity
(a and b) and plastic strain
intensity (c and d) (V.
Filanovsky field)
1.07
1.27
1.08
1.44
1.49
9.2 Calculation of Zone of Inelastic Deformation in Case …
151
Fig. 9.14 Stress intensity calculated with (a) and without (a) accounting for anisotropy of elastic, plastic and filtration
properties (Kainsaiskoye field)
Fig. 9.15 Intensity of plastic strain calculated with (a) and without (b) accounting for anisotropy of elastic, plastic,
filtration properties (Kainsaiskoye field)
Fig. 9.16 Intensity of stress (a), and plastic strains (b), calculated with account for the anisotropy of elastic, plastic and
filtration properties for the perforation cut located along the normal to layering (Kainsaiskoye field)
152
9 Mathematical Modeling of Mechanical and Filtration Processes …
Fig. 9.17 Pore pressure calculated with account for the anisotropy of elastic, plastic and filtration properties for the cut
located along (a) and normally to (b) bedding (Kainsaiskoye field)
References
Karev VI, Kovalenko YuF, Zhuravlev AB, Ustinov KB
(2015) Filtering model into a well taking into account
the permeability dependence on the stresses. Processy
v geosredah (Processes in GeoMedia) 4(4):35–44 (in
Russian)
Karev VI, Kovalenko YuF, Sidorin YuV, Ustinov KB
(2016) Geomechanical modeling of processes in
bottom-hole zone. Monitoring. Nauka i tehnologii
(Monit Sci Technol) 3(28):85–91 (in Russian)
Karev VI, Kovalenko YuF, Ustinov KB (2017a) Modeling deformation and failure of anisotropic rocks
nearby a horizontal well. J Min Sci 53(3):425–432.
https://doi.org/10.1134/s1062739117032319
Karev VI, Kovalenko YuF, Sidorin YuV, Stepanova EV,
Ustinov KB (2017b) Modeling of fluid seapage in a
well at great depths accounting the anisotropy of
reservoir strength properties. Processy v geosredah
(Processes in GeoMedia) 2(11):512–521 (in Russian)
Karev VI, Klimov DM, Kovalenko YuF, Ustinov KB
(2018a) Modelling of mechanical and filtration processes near a well with regard to anisotropy. J Phys:
Conf Series 991:012039. https://doi.org/10.1088/
1742-6596/991/1/012039
Karev VI, Klimov DM, Kovalenko YuF (2018b) Modeling geomechanical processes in oil and gas reservoirs
at the true triaxial loading apparatus. Phys Math
Model Earth Environ Process 30:336–349. https://doi.
org/10.1007/978-3-319-77788-7_35 (Ser. Springer
Geology)
Karev VI, Klimov DM, Kovalenko YuF, Ustinov KB
(2018c) Modeling of deformation and filtration processes near wells with emphasis of their coupling and
effects caused by anisotropy Phys Math Model Earth
Environ Process 30:350–360 https://doi.org/10.1007/
2f978-3-319-77788-7_36 (Ser. Springer Geology)
Karev VI, Klimov DM, Kovalenko YuF, Ustinov KB
(2018d) Modeling of deformation and filtration processes near producing wells: Influence of stress state
and anisotropy. In: Litvinenko (ed) Proceedings of
symposium EUROCK2018. Geomechanics and geodynamics of rock masses. Taylor & Francis Group,
London. 2:1381–1386
Karev VI, Klimov DM, Kovalenko YuF, Ustinov KB
(2018e) Physical modeling of real geomechanical
processes by true triaxial apparatus. In: Litvinenko
(ed) Proceedings of symposium EUROCK2018.
Geomechanics and geodynamics of rock masses.
Taylor & Francis Group, London. 2:1375–1380
Klimov DM, Karev VI, Kovalenko YuF, Ustinov KB
(2009) On the stability of inclined and horizontal oil
and gas wells. In collection: “Actual problems of
mechanics. Mechanics of solid”). Ed. In chief Goldstein R.V. Ishlinsky Institute for problems in Mechanics RAS. Nauka, Moscow, 520p, pp 455–469. ISBN
978-5-02-036961-0 (in Russian)
Klimov DM, Karev VI, Kovalenko YuF, Ustinov KB
(2015) Interaction of stress-strain state and filtration in
rocks. In collection: “Actual problems of mechanics:
50 years of the Ishlinsky Institute for Problems in
Mechanics RAS”). Ed. In chief Chernousko F.L.
Ishlinsky Institute for Problems in Mechanics RAS.
Nauka, Moscow, 510p, pp 489–508. ISBN
978-5-02-039181-9 (in Russian)
Kovalenko YuF, Ustinov KB, Zhuravlev AB (2016)
Stress-strain state in the vicinity of a perforated well
taking into account inelastic deformation. Processy v
geosredah (Processes in GeoMedia) 1(5):69–76 (in
Russian)
References
Ustinov KB (2016) On application of models of plastic
flow to description of inelastic behavior of anisotropic
rocks. Processy v geosredah (Processes in GeoMedia)
3(7):278–287 (in Russian)
Zhuravlev AB, Karev VI, Kovalenko YuF, Sidorin YuV,
Sirotin AA, Ustinov KB (2012) On plastic deformation of rocks. Determination of plastic characteristics
according to experiments on triaxial loading. In:
Collection of papers of the 3d International Confer-
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Ocober 8–12, 2012 (Tsakhkadzor, Armenia), pp 238–
242. ISBN 978-9939-63-129-5 (in Russian)
Zhuravlev AB, Karev VI, Kovalenko YuF, Ustinov KB
(2014) The effect of seepage on the stress-strain state
of rock near a borehole. J Appl Math Mech 78(1):56–
64
Directional Unloading
Method is a New Approach
to Enhancing Oil and Gas Well
Productivity
10.1
Technology of Directional
Unloading a Reservoir
Numerous studies of core material on TILTS
allowed developing a new method of enhancing
the productivity of oil and gas wells—the method
of directional unloading of a reservoir. The basis
of this method is the revealed phenomenon of
increase of rock permeability due to their
cracking and destruction under the influence of
stresses of a certain kind and level.
One of the main reasons for the decrease in oil
and gas well production rate is a decline of rock
permeability in the bottom-hole zone. It happens
both at the stage of well drilling due to penetration of drilling mud into the formation and during
the process of well operation due to silting filtration channels. Correspondingly, certain methods of controlling the decline of rock
permeability in the bottom-hole zone are applied
both in the process of drilling of wells and in the
course of their operation (well workover).
When drilling wells, various methods are used
to prevent the penetration of drilling mud particles into the formation. Polymer-based drilling
fluids can be used for this purpose. Another
method is drilling in equilibrium, i.e. when the
mud pressure on the bottom-hole is kept equal to
the formation fluid pressure. Nowadays, underbalanced drilling is used, when the bottom-hole
pressure is maintained below the oil formation
pressure by 30–40%. These technologies have an
effect, although not always, however they are
© Springer Nature Switzerland AG 2020
V. Karev et al., Geomechanics of Oil and Gas Wells,
Advances in Oil and Gas Exploration & Production,
https://doi.org/10.1007/978-3-030-26608-0_10
10
extremely expensive, especially underbalanced
drilling. For implementation of these methods it
is necessary to have specialized expensive
equipment, the process of drilling is considerably
lengthened and complicated.
During well workover, almost all the methods
currently in use are aimed at improving the permeability of the bottom-hole zone by “cleaning”
the clogged filtration channels during operation
(as well as during drilling). These are acid
treatment, hydro impulse methods, acoustic
methods, vibration methods, methods of alternation of short-term repression and depression,
etc. Acid treatment of wells is most widely used,
which is apparently due to its cheapness and
simplicity, although the efficiency of its use,
particularly in Western Siberia, is not high.
Special mention should be made of the
method of hydraulic fracturing. This method is
aimed at creating a large surface area of oil filtration (hydraulic fracturing crack surface) rather
than at bottom-hole zone treatment. Hydraulic
fracturing is currently the most effective way of
well workover. Its main disadvantage is a high
price and the need to use special equipment and
materials. In addition, hydraulic fracturing is
difficult for deep fields (3 km and more).
Now there is a practice, when already at the
stage of well completion some measures are
taken to restore the permeability of the
bottom-hole zone. For this purpose, acid treatment is most often used. So in particular, on the
fields of the Perm region of the Astrakhan region,
155
156
10
after drilling, the well is completed by swabbing
and one or more acid treatments are performed at
once.
A new way to improve the productivity of oil
and gas wells—the method of directional
unloading of the formation (the original name—
the method of geoloosening) has been developed
in the Institute for Problems in Mechanics of the
Russian Academy of Sciences. It is based on the
ideas expressed by Academician S.A. Khristianovich concerning the decisive influence of
the stresses acting in the vicinity of wells on the
filtration properties of rocks and, as a consequence, on the flow rate of oil and gas into wells.
A decline of bottom-hole zone permeability
occurs under almost any conditions of well
construction completion and depends on various
factors. As noted, it is traditionally believed that
the main reason is the impurity of the
bottom-hole zone as a result of the penetration of
filtrate and the solid phase of drilling mud. At the
same time, the effect of stresses on filtration
properties of bottom-hole zone has been studied
quite insufficiently.
Theoretical studies, numerous laboratory tests
of core material on TILTS and pilot and field
operations on wells performed by specialists of
the Institute for Problems in Mechanics of RAS
in recent years have shown that the stresses can
significantly (several times and even dozens of
times) and, which is important, irreversibly
change the permeability of rocks in bottom-hole
zones depending on the structure and deformation properties of the rock, the depth of formation
and formation fluid pressure, construction of well
bottom-hole, and conditions of well operation.
Moreover, the permeability may both increase
and decrease.
The reason for irreversible reduction of permeability of the rock is related to the fact that
when the stresses reach some critical value for
the given material (yield strength), the plastic
deformation of clay contained in the sandstone
begins, which leads to the occlusion of part of
filtration channels. Due to irreversibility of
plastic deformations, the decrease in permeability
is also irreversible, which is observed in practice.
Directional Unloading Method is a New Approach …
The reverse process is also possible. As
studies at TILTS on rock specimens from reservoirs of numerous fields have shown, for the
majority of rocks there are stress states at which
process of fracturing starts to develop leading to
sharp increase in permeability. If these stresses
are implemented in the bottom-hole zone, the
appearing cracks will play the role of new filtration channels, which will lead to a sharp irreversible increase in the permeability in the
vicinity of the well.
These issues had been discussed in more
detail in Chap. 6.
It follows from the above that in order to
develop optimal modes of well completion and
operation it is important to know what consequences from the point of view of changes in
permeability of the reservoir the stresses arising
in rock result to, and what pressure drawdowns
need to be maintained at the bottom-hole in order
to prevent negative deformation processes in the
reservoir. A proper understanding of these processes, the ability to adequately recreate them in
the laboratory conditions and to carry out competent processing of the results obtained gives
the basis for creating new ways to improve well
productivity and increase oil recovery.
Academician S. A. Khristianovich proposed
to use the discovered effect of increasing permeability of rocks by creating the necessary
stress states to increase the permeability of
bottom-hole formation zone, and on the basis of
this idea a new method of increasing the productivity of oil and gas wells—the method of
directional unloading of a reservoir (DUR
method)—was developed.
The essence of the method is to create such
stresses in the vicinity of the well, which result in
rock fracturing and creation of an artificial system of multiple macro-cracks. The permeability
of this system of artificial filtration channels
significantly exceeds the natural permeability of
the reservoir.
Figure 10.1 shows a rock specimen after
hollow cylinder test (Sect. 5.3) which simulates
pressure drawdown on the bottom-hole of an
uncased well. It clearly shows how a system of
10.1
Technology of Directional Unloading a Reservoir
Fig. 10.1 Crack formation around a hole simulating a
well in a strong rock
Fig. 10.2 Creation of a crack zone around a hole
simulating a well in a weak rock
macro-cracks is formed around the holes. Less
strong rocks may disintegrate, turn into sand and
fall out into the hole, Fig. 10.2.
The figures clearly show the formation of
macro cracks, which in the case of wells will
form an artificial mesh of filtration channels over
the silted natural system of filtration channels.
Figures 6.5 and 6.19 show the change in permeability of rock specimens from the Symoriakhskoye oil field and the Astrakhan gas
condensate field during modeling pressure
157
drawdown on the bottom of uncased borehole on
TILTS. It can be seen that at the beginning the
relative permeability of specimens k=k0 (k0 —
initial permeability of the specimen) decreases
with pressure drawdown, and then at some value
of pressure drawdown it sharply increases and
becomes much larger than the initial one.
The second important point of the method of
directional unloading is the need to maintain the
required pressure on the bottom-hole for a sufficiently long time, as the process of the cracks
growth develops gradually, spreading over time
into the reservoir. This is due, firstly, to the
rearrangement of the pressure drawdown funnel
in the vicinity of the well, and, secondly, to the
fact that at high stresses the rocks cease to be
elastic and begin to creep.
Experience of practical implementation of the
DUR method has shown that only a decrease in
bottom-hole pressure does not always result in
stress states in the reservoir required to rock
fracture. Therefore, in some cases before the
bottom-hole pressure reducing it is necessary to
introduce stress concentrators into the reservoir
rock. Such stress concentrators can be perforation holes, vertical or horizontal cuts. The presence of stress concentrators allows not only to
initiate the process of cracks growth in the
vicinity of the well, but also to make it much
more intensive and expanded.
Of course, the easiest thing to do is to make
extra perforation. However, the problem is that
the stresses around the perforation holes depend
largely on their shape and volume, perforation
density, etc. The most suitable perforation holes
for this purpose are those that are close to a
cylindrical shape.
The following five figures demonstrate
schematically the process of cracks formation and
growth in the vicinity of an additional perforation
hole during lowering bottom-hole pressure.
Figure 10.3 shows a cased wellbore section
with a production perforation hole.
The well is additionally perforated. Figure 10.4 shows one of the additional perforation
holes, the shape of which differs significantly
from the operational one.
158
10
Fig. 10.3 Schematic presentation of a wellbore section
with a perforation hole
After additional perforation, the technology
provides for a reduction in bottom-hole pressure.
At some pressure value new cracks starts to grow
around the additional perforation holes,
Fig. 10.5.
In order to make the process of cracks growth
more intensive and to spread it as far as possible
into the reservoir, the bottom-hole pressure is
further reduced and maintained for the required
time. Figure 10.6 illustrates schematically how
the cracks zone increases and captures production perforation holes, which also increases the
flow rate. After performing works by the method
of directional unloading the pressure on the well
bottom-hole is increased to the operational values, Fig. 10.7.
The following should be noted. Pressure in
the well reduces at hydraulic fracturing after the
Fig. 10.4 Additional well
perforation
Directional Unloading Method is a New Approach …
work is done. Correspondingly, the pressure in
the fracturing crack is also reduced. Therefore,
proppant has to be pumped into the crack to
maintain its opening. When using the DUR
method, the bottom-hole pressure increases after
the work is done and created cracks expand
further.
The effect of application of the method of
directional unloading is composed of two factors
—the elimination of the effect of mudding and
the actual increase in the filtration surface around
the well.
As noted above, a zone of reduced permeability is formed near the bottom-hole (zone of
mudding) when drilling and cementing. There
are several reasons for its formation: appearance
of clay crust on the surface of the well, clogging
of natural filtration channels with solid particles
of drilling mud during drilling and with particles
of plugging mud during cementing of wells, etc.
The permeability of the rock in the colmatation
zone can be dozen times lower than natural one.
Deterioration of permeability in the bottom-hole
zone occurs not only when drilling wells, but
also during their operation. As a result, the flow
rate is significantly lower than potentially
possible.
Figure 10.8 shows schematically the vertical
cross-section of the well of radius Rw, surrounded by the zone of reduced permeability
(zone of mudding) of radius R*.
The reservoir pressure distribution in this case
is (Leibenzon 1947) at r < R*
10.1
Technology of Directional Unloading a Reservoir
159
pðrÞ ¼ po po pw
Rc
ln
ð10:2Þ
ko ðk11 ln RRw þ k1o ln RRc Þ r
where Rc is the radius of the supply contour; po is
formation pressure; pw is pressure in the well; k0
is the natural permeability of the reservoir (at
r > R*), k1 is the permeability of the rock in the
zone of mudding (k1 < k0 ).
The flow rate of the unit of well length is
Q1 ¼
Fig. 10.5 The beginning of cracks formation in the rock
when the bottom-hole pressure drops
2pk1 po pw
l ln RRc þ kko ln RR
1
ð10:3Þ
w
where l is the viscosity of fluid.
In the case of the absence of a zone of mudding, the steady flow rate in the well is given by
Dupuit formula (Landau and Lifshitz 1976)
Qo ¼
2pko po pw
l
ln RRc
ð10:4Þ
w
Then the decline of the flow rate in the well is
ln RRc þ ln RRw
Q1
¼
Qo ln RRc þ kko ln RR
1
w
Fig. 10.6 Growth of the crack formation zone with
further decrease of bottom-hole pressure
Finally, if a ¼ ln RRw and b ¼ ln RRc , then
1 þ ab
Q1
¼
Qo 1 þ ab kko
1
ko
k1
Fig. 10.7 Artificial crack system in the vicinity of the
well after the DUR works
pðrÞ ¼ pw þ
at r > R*
k1 ðk11
po pw
r
ð10:1Þ
Rc ln R
R
1
ln Rw þ ko ln R Þ
w
ð10:5Þ
ð10:6Þ
If Rw = 0.1 m, Rc = 250 m and R* = 0.2 m,
= 10, then the well flow rate will decrease by
1.8 times, if kko1 = 50, the well flow rate will
decrease by 5.5 times.
2. The second factor leading to an increase in
the well flow rate when using the method of
directional unloading is the actual increase in the
surface area of filtration due to a significant
increase in the permeability of the bottom-hole
formation zone. Figure 10.9 shows schematically
a section of the well in the productive part of the
reservoir, and the zone (shaded) in which the
crack formation occurred. If the permeability in
the cracks zone k2 significantly exceeds the
160
10
Directional Unloading Method is a New Approach …
Fig. 10.8 Schematic representation of the bottom-hole
formation zone
Fig. 10.9 Schematic representation of the high permeability zone
natural permeability of the reservoir rock, then
for filtering fluid, this is the same as formation of
a “cavity” around the well having the same shape
and size as the cracked zone. The actual surface
of fluid filtration from the reservoir increases and
the flow rate of the well increases proportionally.
The determination of the shape and size of the
“cavity” is a complex three-dimensional mathematical problem, moreover it is needed to be
carried out additional tests of reservoir rock.
It should be emphasized once again that the
cracks zone around the well can be considered as
a “cavity” only in terms of fluid filtration from
the reservoir, because it provides very little
resistance to the flow of the fluid due to its high
permeability. In fact, it is, of course, the rock, but
the rock much more cracked and decompacted
than the natural reservoir rock.
So to develop optimal modes of well completion and operation it is important to know
what consequences stresses increase results in
changing of permeability, and what pressure
values need to be maintained on the bottom-hole,
in order, on the one hand, to prevent negative
deformation processes in the formation, and, on
the other hand, to initiate the process of cracking
in the vicinity of the well, thereby increasing the
permeability of the rock in this area.
To answer this question, the same approach
was used as to solving the problem of wellbore
stability. The first stage is to calculate the stresses
acting in the well vicinity at various bottom-hole
designs and their change with the bottom-hole
pressure change. In simple cases (open wellbore)
there are analytical solutions, in more complex
cases (casing, perforation holes, cuts, etc.)
numerical methods are implemented by using
three-dimensional programs to calculate the
stress-strain state. Each of the above mentioned
cases of bottom-hole design has its own program
of specimen loading which corresponds to
gradual decrease in bottom-hole pressure.
Then the analyzed situation is directly simulated on TILTS. For this purpose, the calculated
stresses are applied to the rock specimen and the
specimen strains in three directions and permeability in one direction are measured. As a result,
stresses and, accordingly, bottom-hole pressures
are determined at which the process of cracking
or destruction begins in the reservoir. On the
10.1
Technology of Directional Unloading a Reservoir
161
basis of these data, a plan is drawn up for the
implementation of the method of directional
unloading for a particular well.
characterized not by two, but by five constants of
elasticity.
2. Calculation of stresses in the bottom-hole
zone for various bottom-hole designs.
10.2
Methodology for Well
Productivity Enhancing
by Means of Directional
Unloading
The development of the technological regulations
for enhancing well productivity by the method of
directional unloading (DUR) at a particular field
includes the following stages.
1. Testing of core material from the reservoir of
the field under study on the experimental
stand of TILTS.
One of the key points of the method of
directional unloading of the reservoir is the
determination of the type and level of stress at
which the process of cracking begins in the
bottom-hole part of the reservoir. Obviously, the
values of these stresses and their type will be
different for various rocks, reservoir conditions,
formation pressure and a number of other factors.
And they can only be determined experimentally
by true triaxial testing core material from the
field under study.
As a result of testing rock specimens at the
TILTS should be determined:
– the stress values that need to be created in the
bottom-hole zone in order to cause the process
of micro- and macro-cracking or destruction of
the rock, accompanied by an irreversible
increase in its permeability;
– elastic constants of rock required to calculate
the stress-strain state in the bottom-hole zone
at various bottom-hole designs (open bore,
casing, type of perforation, oriented slots, etc.).
The rocks composing the reservoirs of oil and
gas fields, primarily, sandstones have a pronounced layered structure. Therefore, their strain
and strength properties are close to those of a
transversally isotropic material and are
Calculations at the second stage of the adaptation of the method of directional unloading to
the conditions of a particular field should answer
the question how to create in the vicinity of the
well the stresses determined during rock testing
at the first stage.
In fact, there are two possibilities to change
the existing stresses in the formation:
bottom-hole pressure control and creation of the
necessary bottom-hole design.
During the calculations it is necessary to find
out whether it is possible to initiate the process of
cracking in the vicinity of the well for this
bottom-hole design (casing, filter-shank, perforation type, etc.). Depending on whether the well
is cased or not, there will be completely different
stress states in its vicinity at the same pressures
on the bottom-hole. If it turns out that this well
design does not allow to initiate the process of
rock fracturing even with the maximum pressure
drawdown on the bottom-hole, then the question
arises what technological measures should be
taken to make it possible.
Firstly, whether the perforation should be
cumulative or slotted. If cumulative perforation is
required, a number of questions should be
answered: what should be the diameter of the
holes and their length; what should be the density
of the perforation; what intervals should the
additional perforation be performed in.
When selecting the slotting perforation, it is
necessary to determine the direction of the slots
—horizontal or vertical. For the selected type of
perforation it is necessary to determine the level
of pressure drawdown at the bottom-hole to initiate the process of crack formation. There are
other factors that need to be taken into account in
the calculations.
Answering to the above questions is a complex problem, because they require numerical
solutions to essentially three-dimensional problems of elasto-plasticity and fracture.
162
10
Figures 10.10, 10.11, and 10.12 depict calculations of stress fields for three typical
bottom-hole design: a cone-shaped perforation
hole in a cased well (Fig. 10.10); a cone-shaped
perforation hole in a uncased well (Fig. 10.11);
two cone-shaped perforation holes in a cased
well (Fig. 10.12).
Each of the figures shows the isolines of the
intensities of shear stresses responsible for rock
fracture. The isolines, represented in fractions of
rock pressure, correspond to the maximum
pressure draw down at the bottom-hole. Calculations of basic problems were carried out for
isotropic media with some elastic constants
characteristic of rocks.
3. Drawing up the technological regulations of
work by the method of directional unloading
on wells of a particular field.
Directional Unloading Method is a New Approach …
Technological regulations include preliminary
technological operations on wells and selection
of technical and technological parameters of well
treatment using DUR method.
The advantages of the directional unloading
method are as follows.
1. Understanding that the state of the
bottom-hole zone has a decisive influence on
well operation has led to the implementation
of measures to maintain or restore permeability in the bottom-hole zone during the
drilling and completion stages.
In the first case, drilling is carried out in
equilibrium or underbalance to prevent the drilling mud from penetrating into the formation and
thus to prevent its filtration properties from
deteriorating. However, as noted above, these
Fig. 10.10 Distribution of shear stress intensities in the vicinity of the cased well with a cone perforation hole
10.2
Methodology for Well Productivity Enhancing …
163
Fig. 10.11 Distribution of shear stress intensities in the vicinity of an uncased well with a conical perforation hole
technologies are extremely expensive and significantly lengthen the drilling process itself.
In the second case, after drilling by using a
weighted mud, the well is completed by one of
the traditional methods (usually swabbing or
compressing), and then immediately at the stage
of completion, the measures to restore the permeability of the bottom-hole zone are carried out
(acid treatment is the most widely used now).
This also significantly increases the cost of well
completion (not to mention the fact that the result
is often insignificant), because the completion
and treatment of the bottom-hole zone requires
different equipment, there is a need for additional
downhole operations, the duration of the well
completion phase lengthens significantly, etc.
Using the method of directional unloading
allows to combine these two operations into one,
i.e. to combine a well completion with simultaneous restoration of permeability in the
bottom-hole zone. There is no need to use any
additional equipment or to carry out additional
lowering and lifting operations. As a result, the
cost and time of a well completion is significantly reduced, while the quality of work is
improved.
2. The method of directional unloading is
applicable to all reservoir depths. Moreover,
its efficiency for deep fields (3 km and more)
might be higher than for shallow ones (less
than 1.5–2 km). The abnormally high reservoir pressure also contributes to the successful application of the directional unloading
method.
Practice shows that 2–4 times increase in flow
rate is usually achieved on uncased boreholes
and 1.5–2 times increase—on cased boreholes.
The duration of the effect is usually from several
months to a year.
3. The implementation of DUR method requires
standard equipment available at every field.
164
10
Directional Unloading Method is a New Approach …
Fig. 10.12 Distribution of shear stress intensities in the vicinity of the cased well with two cone perforation holes
The developed technology is protected by 7
Russian patents and 1 Eurasian patent (Khristianovich et al. 1998; Kovalenko et al. 2001,
2002a, b, 2003a, b; Karev et al. 2006).
10.3
Practical Implementation
of the Directional Unloading
Method
The directional unloading method was successfully applied at a number of fields in Western
Siberia and the Perm region for well completion,
workovers of producing and injection wells.
In the course of the work, a pressure drawdown of the required level and duration is created
at the bottom-hole by using a jet pump. The main
parameters at the bottom-hole (pressure, temperature, flow rate) are controlled by using a
multipurpose geophysical device and geophysical station. Figure 10.13 shows the layout of the
equipment, lowered into the well during the work
on DUR method.
Injection of the working fluid as which technical water or technical oil can be used is carried
out by a pumping unit. A special insert with a
multipurpose geophysical device connected to its
lower end is lowered into the body of the jet
pump by using the geophysical cable. It is connected to a geophysical station on the earth’s
surface by means of an electrical wire running
inside the special insert and a geophysical cable.
The well is treated by the jet pump for a certain
period of time, also a cyclic effect on a reservoir
is possible by alternating switching the pump
unit on and off.
After such well treatment, it is advisable to
conduct hydrodynamic studies of the well and
record the pressure recovery curve in order to
assess the efficiency of the impact. For this purpose, another special insert with a self-contained
pressure gauge attached to it is lowered into the
10.3
Practical Implementation of the Directional Unloading Method
165
Fig. 10.13 Equipment
layout for oil well directional
unloading method
jet pump housing. This insert is equipped with a
check valve that prevents the working fluid from
flowing into the well section under the packer.
Thus, when pumping through the jet pump, the
valve is open and a pressure draw down is created at the bottom-hole, when pumping stops, it
closes, and the fluid can only come into the space
under a packer from the reservoir. The hydrodynamic characteristics of the well are determined by the pressure recovery curve.
It is advisable to combine the technology of
directional unloading with such a widespread
enhanced oil recovery method as acid treatment
or bottom-hole zone treatment by means of other
chemical agents. Preliminary directional unloading operation significantly increase permeability
of bottom-hole zone, which is usually lower than
the natural permeability of the reservoir, thus
allowing faster and deeper penetration of the
reagent into the reservoir. Spent substance is
pumped out of the reservoir by means of a jet
pump.
Gas wells do not require the use of jet pumps,
creation of pressure drawdown of the required
value is carried out by installation of fittings of
the appropriate diameter at the wellhead.
166
10
Below are some results obtained from the
application of directional unloading technology
on the wells.
Uncased borehole:
“LUKOIL—Western Siberia”, Symoryakhskoye
field, a producing well No. 7197, a completion:
expected flow rate—6 tpd, received—24 tpd;
a producing well No. 7197, workover: before—
3 m3/day, after-9 m3/day.
“Slavneft”, Novo-Pokurskoye field, a producing
well 99, workover:
before—2 m3/day, after—8 m3/day.
Cased borehole:
“RITEK”, Kislorskoye field, a producing well
302, workover:
before—4 m3/day, after—9 m3/day;
a producing well 303, workover: before—
5 m3/day, after—9 m3/day;
a producing well 331, workover directional
unloading + acid treatment: before—6 m3/day,
after—11 m3/day;
“LUKOIL—Perm”, Siberian field, injection well
310, workover: before—8 m3/day, after—
200 m3/day;
well 310, second workover: before—5 m3/day,
after—100 m3/day;
a producing well 301, workover: before—
6 m3/day, after—90 m3/day;
a producing well 338, workover: before—
3 m3/day, after—9 m3/day.
The directional unloading method can be
applied to any type of field. The expected effect is
Directional Unloading Method is a New Approach …
estimated based on the results of core material
testing and related calculations. Practice shows
that 2–4 times increase in flow rate is usually
achieved on uncased boreholes and 1.5–2 times
increase on cased boreholes. The duration of the
effect is usually several months—up to a year.
References
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gases in porous medium. M.-L.: Gostekhizdat, 244p
(in Russian)
Landau LD, Lifshitz EM (1976) Statistical physics. Part 1:
Edition 3, supplemented. M.: Science, 584p (in
Russian)
Karev VI, Kovalenko YuF (2006) Dependence of the
bottom-hole formation zone permeability on the
pressure drawdown and the bottom-hole design for
different types of rocks. In: Technologies of the fuel
and energy complex, 6:59–63 (in Russian)
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(1998) A method of completing. The patent of the
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well completion method. The patent of the Russian
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Kovalenko YuF, Kulinich YuV, Karev VI et al (2002b) A
workover method of wells. The patent of the Russian
Federation No. 2188317, 27.08.2002
Kovalenko YuF, Kulinich YuV, Karev VI et al (2003a) A
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