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1. INTRODUCTION
Gas hydrates are metastable crystalline materials
consisting of one or more type of gas molecules inside amolecular cage made out of hydrogen bonded water
molecules. The gas hydrates form because of existence
of hydrogen bonds between water molecules as well as
Vander Waals forces between water and gas molecules
[1]. The stability of gas hydrates compounds relates to
parameters like temperature, pressure, gas composition
and even external electrical and magnetic fields [2, 3].
Gas hydrate prospects on earth are the biggest unlocked
recourses of energy in the world. The estimated amount
of organic carbon in the form of gas hydrate in earth is
10000 giga tones which is equal to 105
to 3109
Tcf of
gas reserves [4, 5]. The significance of gas hydrateresources will be recognized when we consider that the
total amount of non hydrate gas reservoirs in the world is
just 13000 Tcf [5]. A proposed method for gas
production from gas hydrate reservoirs is
depressurization. For production from gas hydrate
reservoirs by depressurization, a production model that
could predict the behavior of gas hydrate reservoir
during production is absolutely essential. The knowledge
of stress and strain distribution around the wellbore
during production is needed to assess the problems like
wellbore stability and potential for sand production.
Different researchers addressed the problem of stress and
strain distribution around wellbore with different
approaches. Freij-Ayoub et al [6] used FLAC to
numerically calculate stress and strain distributionaround the wellbore induced drilling through gas hydrate
bearing strata. Rutqvist el [8] used TOUGH+HYDRATE
to numerically simulate pressure and temperature
distribution around the wellbore induced by different
thermal and mechanical conditions during gas hydrate
dissociation. Then, FLAC3D was used to calculate stress
distribution around the wellbore. Kimoto et al. [8], on theother hand, treated hydrate bearing reservoir as a chemo
thermomechanical material and used an elasto
viscoplastic model to address plastic deformations in the
soil during gas hydrate production. However, the selection
of appropriate model to simulate the condition of stress andstrain around the wellbore is greatly affected by the
geology of reservoir as well as the condition of production
from the reservoir. Waite et al pointed out that gas hydrate
accumulations in coarse grain sands are more prone to
plastic deformation and sand production during production
period than fine grain hydrate sediments [9]. The selection
between a poroelastic or poroplastic model is related to the
gas production rate and whether the gas hydrates bear loads
in the reservoir or merely fill the voids in the pore space. In
this paper, it is assumed that the reservoir remains elastic
during gas production period. In addition, the intrinsic
ARMA 11-540
Stresses around a Production Well in Gas Hydrate-Bearing Formation
Sadegh Badakhshan Raz
Harold Vance Department of Petroleum Engineering, Texas A&M University, College Station, TX, USA
Ahmad Ghassemi
Harold Vance Department of Petroleum Engineering, Texas A&M University, College Station, TX, USA
Copyright 2011 ARMA, American Rock Mechanics Association
This paper was prepared for presentation at the 45th
US Rock Mechanics / Geomechanics Symposium held in San Francisco,CA, June 2629, 2011.
This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technicaland critical review of the paper by a minimum of two technical reviewers. The material, as presented, does not necessarilyreflect any position of ARMA, its officers, or members. Electronic reproduction, distribution, or storage of any part of thispaper for commercial purposes without the written consent of ARMA is prohibited. Permission to reproduce in print isrestricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuousacknowledgement of where and by whom the paper was presented.
ABSTRACT:This paper presents an analytical solution for the temperature, pore pressure, strain and stress distributions around awellbore in a gas hydrate reservoir during gas production by depressurization. The problem of decomposition of gas
hydrate in porous rock is treated as a Stefan type moving boundary problem. The pressure distribution as well as
temperature distribution is obtained with solution of gas diffusivity equations and conductive-convective heattransfer equations. These are then used in a displacement function in terms of the pore pressure and temperature
distributions to find the resulting stress distributions in the rock. A number of simulations are presented to highlight
the impact of production on the pore pressure and stress. The results of this semi-analytical study can be used tobenchmark more complex numerical methods for the wellbore stability analysis in gas hydrate formations.
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permeability of rock remains the same despite the
change in gas effective permeability in non decomposed
and decomposed zones. It is assumed that the fluid flow
is single phase flow i.e. gas flow, and water remains
stagnant in the reservoir [3]. A poroelstic model is used
as a semi-analytical method to calculate the induced total
stress in gas hydrate reservoir during gas production.
2. MATHEMATICAL MODELS
For calculation of induced stress and strain in gas
hydrate reservoir during production, we need to know
the pressure and temperature distribution in the reservoir
during decomposition of hydrate layer. The area around
a wellbore can be viewed as two parts, one
corresponding to the decomposed region and the other
the non-decomposed gas hydrate layers. The schematic
of these layers is shown in figure 1.
Fig. 1. Different zones during gas production, rw is wellbore
radius, R is the radius of gas hydrate decomposing front and re
is reservoir radius
During gas production, the diameter of decomposed
layer, R, will grow with time. The movement of the
boundary between the two areas introduces a physical
problem with free moving boundary condition which is
called Stefan problem named after Joef Stefan, the
Slovene physicist who studied the problems of ice
formation around 1890[10]. Stafans problems include at
least two differential equations with their own boundary
conditions which are related together through Stefans
condition. The mathematical model of gas hydrate
production as well as the resulted strain and stress fieldaround the wellbore are discussed in following sections.
In the following section, we elaborate on the method
introduced by Makogon et al [3] to calculate temperature
and pressure distribution during gas production from gas
hydrate bearing reservoir.
2.1. PRESSURE DISTRIBUSIONThe governing equation for pressure distribution around
the wellbore during gas hydrate production is gas
diffusivity equation in polar coordinates [3, 11]:
(1)
Where n is equals 1 for decomposed gas hydrate layer
zone and is 2 for non decomposed zone. Also P is the
pore pressure and k is permeability to gas, is gas
viscosity, s is water saturation, is hydrate saturation,
phi is porosity in both decomposed and non-decomposed
layer.
In addition,
(1-s)m
m2 = (1-) m
The above equation is non linear and can be linearized
with respect to P in order to be solved analytically. For
the linearization of gas diffusivity equation, we consider
following approximations [3].
(2)
(3)
Where Pe is the reservoir initial pressure in MPa and PD,
MPa, is equilibrium pressure at the interface between
decomposed and non decomposed layers. After
linearization, we will obtain the following equations:
(4)
Where:
(5)
(6)
The boundary conditions for diffusivity equations are:
P(rw , t) = PW (7)P2(r, 0) = P2 (, t) =Pe (8)P1(R(t), t) = P2 (R(t), t) =PD (9)
Where rw is wellbore radius in meter, PW is wellbore
pressure, R(t) is the radius of the interface between the
decomposed and un-decomposed layer. The solutions of
linearized gas diffusivity equation are [3, 11]:
(10)
(11)
The functions and coefficients in equations 10 and 11 are
[3]:
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presented. The main assumption in this study is that the
gas production rate is constant. Other assumption is that
mechanical behavior of reservoir remains in elastic
region. For coefficients and mentioned parameters in the
paper, the hypothetical case of gas hydrate reservoir is
considered with the parameters same as those in Ji et al.
[11] and Freji-Ayoub [9]. These parameters are
mentioned in table 1.
Table1. The amount of parameters used in this
study
In this study, the effect of production time and gas
production rate on pressure, temperature, strain and
stress distribution in reservoir were investigated. The
wellbore radius, rw, is assumed to be 0.13m. For all the
calculations, the initial reservoir pressure, Pi, and initial
reservoir temperature, Ti, are considered to be 15 MPa
and 287 K, respectively. Also, the permeability in
decomposed layer is k1= 5.210-15
m2
(5.2 md) and in
the non decomposed hydrate layer is k2= 0.410-15 m2
(0.4 md). Other gas properties like viscosity and
permeability are assumed to be constant. This seems to
be reasonable assumption for low temperature and fairly
low pressure gas hydrate reservoirs. A Mathematica
code was written to do all the necessary calculation.
3.1 The EFFECT OF PRODUCTION TIME
Consider the constant gas production rate of 0.04
kg/second, production times are considered to be 120,
365, 100 and 730, 1460 and 2920 days (0.33, 1, 2, 4 and
8 years, respectively). The effect of production time on
pressure and temperature around the wellbore are shown
in Figures 2 and 3. As it is shown in Figure 2, there are
two different zones of pressure in the gas hydrate on
either sides of gas hydrate decomposing front. The position of the decomposition front coincides with a
bump in pressure and temperature graphs. The bump is
caused by considerable permeability difference on either
sides of the front. The lower permeability in the non-
decomposed gas hydrate layer corresponds to less
pressure drop at constant production rate. In contrast, the
considerable higher permeability in the decomposed
layer caused sharper pressure drop.
Fig. 2. The effect of production time on pressure
distribution around the wellbore
Figure 3 shows the temperature distribution in the
reservoir in different production times. Like pressure
graphs in figure 2, there is a temperature bump at the
location of decomposition front because of this fact that
two different partial differential equations stated at
equation 18 represents the physics of problem at either
sides of the front. In the decomposed layer near the
wellbore, there is more pressure drops and therefore
Parameter Amount Ref.
: Biots coefficient 0.79 8
: The thermal expansion
coefficient of rock, K-1
1.2 10-5 9
: hydrate saturation, % 0.15 3
: throttling coefficient of gas
(K/Pa)
8 10-7 3
: adiabatic coefficient of gas
(K/Pa)
3.2 10-6 3
: gas viscosity, Pa.s 1.5 10-5
3
: Poissons ratio 0.2 8
: water saturation, % 0.15 3
c1: heat capacity of zone 1
(J/K.kg)
2400.2 3
c2: heat capacity of zone 2
(J/K.kg)
1030.2 3
cv: volume heat capacity of
gas (J/K.kg)
3000 3
0: density of methane gas at
P0 and T0, kg/m30.706 3
3:density of hydrate, kg/m3 0.91 103 3
w: density of water, kg/m3 1 103 3
G: Shear modulus, MPa 6000 8
k1: gas permeability in zone 1,md
5.2 3
K2: gas permeability in zone
2, md
0.4 3
K: Bulk modulus, MPa 8000 8
P0 : atmospheric pressure,MPa
0.101 3
T0 : atmospheric temperature, 273.15 3
m: porosity, % 0.19 3
z: compressibility of gas 0.88 3
Time increase
0 5 10 15 20 25 30 35
284.5
284
283.5
283
282.5
282
2 1. 5
281
280.5
Radius, m
Temperature,
K
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higher velocity which causes higher temperature
decrease due to higher convective heat transfer and
stronger cooling Joule-Thompson effect.
Fig.3. The effect of production time on temperature
distribution around the wellbore
Also like the effect of production time on pressure
distribution, the overall temperature in the reservoir
decreases with increase in production time.
Fig.3. The effect of production time on normal strain
field, around the wellbore
The effect of production time on radial stress is shown in
Figure 3. The type of stress here is compressional stress.
As it is shown in the Figure 3, the radial stress is zero at
the wellbore radius and then increases to a maximum
near the wellbore. The induced radial compressive stressincreases with increase in time due to increase in
induced pressure and temperature. Also, the effect of
production time on tangential stress is shown in Figure
4. The type of induced tangential stress here is
compressive stress. The maximum of induced tangential
stress happens in the wellbore wall and then decreases
with increase in radius. Like induced radial stress,
tangential stress increase with increase in production
time.
Fig.4. The effect of production time on tangential strain
field, around the wellbore
3.2 THE EFFECT OF PRODUCTION RATE
In constant production time of 365 days, the effect of gas
production rate on induced strain and stress distribution
in the reservoir during production period is studied.Different production rates of 0.04, 0.06, 0.08, 0.1 and
0.12 Kg of gas per second are considered. Figure 5,
shows the effect of production rate on induced radial
stress in the reservoir. The type of radial strain is
tensional strain. As it is shown in figure 5, increase in
production rate causes the increase of induced radial
stress due to increase in induced pressure and
temperature in reservoir. The effect of increase in
production rate on induced radial stress is shown in
figure 5. The type stress here is compressional stress.
The induced radial stress is zero in the wellbore and then
increases to a maximum near the wellbore. The inducedradial stress vanishes with increase in radius.
Finally as it is shown in figure 6, the increase in
production rate causes increase in induced tangential
stress in the reservoir.
r/rw
RadialtensilestressMPa
0 5 10 15 20 25 30 35 40
1.2
1
0.8
0.6
0.4
0.2
0
Rate increase
Time increase
0 50 100 150 200 250
r/rw
Tangentialtensilestress,
MPa
0.4
2
1.8
0.8
0.6
1.2
1
1.6
1.4
Time increase
0 50 100 150 200 250
0.9
0.8
0.7
0.6
0.5
0.3
0.2
0.1
0
0.4
r/rw
R
adialtensilestress,
MPa
Time increase
13
12.5
12
11.5
11
10.5
10
9.5
9
8.5
80 5 10 15 20 25 30 35
Radius, m
Pressure,
MPa
Fig.5. The effect of gas production rate on
radial stress field, around the wellbore
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Fig.6. The effect of gas production rate on tangential
strain field, around the wellbore
As it is observed from the radial and tangential stress
results, the highest amount of stresses are induced
around the wellbore. The tangential stresses are
maximum in the wellbore periphery and radial stress
reaches the maximum value near the wellbore becausethere is the maximum pressure drop around the wellbore.
Consequently, the area near the wellbore is the most
prone zone in the reservoir to meet the failure criteria
during gas production.
CONCLUSION
1. The pressure and temperature distribution in gashydrate bearing reservoir during gas production
are calculated analytically
2. The results of pressure and temperaturedistribution are coupled to a linear thermo-
poroelstic model to calculate induced strain and
stress distribution in the reservoir during gas
production.
3. Increase in production time causes an increase ininduced stress and strain in the reservoir
4. Increase in production rate causes an increase ininduced stress and strain in the reservoir
5. The amount of tangential strain and radial stressis zero in the wellbore periphery which increases
to a maximum near the wellbore and againdecreases with increase in radius.
6. The amount of radial strain and tangential stressis maximum in wellbore and decrease with
increase in radius.
REFERENCES
1. Sloan E. D., C.A. Koh, Clathrate Hydrates of NaturalGas, Third Edition, CRC Press, 2009
2. Makogon Yuri F., Stephen A. Holditch, Gas solubilityin water, kinetics and morphology of gas hydrate,
Annual report for Arco Exploration and production
technology, Texas A&M University,1998
3. Makogon Yuri F., Hydrates of Hydrocarbons,1997,Pennwell books, USA
4. http://oceanexplorer.noaa.gov/explorations/deepeast01/background/fire/media/carb_dist.html
5. Editorial, An introduction to natural gashydrate/clathrate: The major organic carbon reserve onearth, Journal of Petroleum Science and Engineering,
Vol. 56, PP 1-8, 2007
6. Reem Freij-Ayoub, Chee Tan, Ben Clennell,Bahman Tohidi, Jinhai Yang, A wellbore stability
model for hydrate bearing sediments, Journal of
Petroleum Science and Engineering 57 (2007)
209220
7. Jonny Rutqvist and George J. Moridis, NumericalStudies on the Geomechanical Stability of
Hydrate-Bearing Sediments, Offshore Technology
Conference, 2007, Houston, U.S.A.
8. Sayuri Kimoto, Fusao Oka, Tomohiko Fushita, Achemothermomechanically coupled analysis of
ground deformation induced by gas hydrate
dissociation, International Journal of Mechanical
Sciences, 52,2010, 365376
9. W. F. Waite, J. C. Santamarina, D. D. Cortes, B.
Dugan, D. N. Espinoza, J. Germaine, J. Jang, J. W.
Jung, T. J. Kneafsey, H. Shin, K. Soga, W. J.
Winters, T.-S. Yun, Physical properties of
hydrate-bearing sediments, Reviews of
Geophysics, Vol. 47,2009
10. L.I. Rubinstein, The Stefan Problem, AmericanMathematical Society,1971
11. Chaung Ji, Goodarz Ahmadi, Duane Smith,Constant rate natural gas production from a wellin a hydrate reservoir, Energy Conversion &
Management, Vol. 44, 2003, PP2403-2423
12. A. Ghassemi, Q. Tao, A. Diek, Influence of coupledchemo-poro-thermoelastic processes on pore pressureand stress distributions around a wellbore in swelling
shale, Journal of Petroleum Science and Engineering 67
(2009) 5764
13. Qingfeng Tao, Wellbore Stability in water sensitiveshales, Master of Science Thesis, University of North
Dakota, 2006
14. Detournay, E. and Cheng, A.H.-D., Fundamentals of poroelasticity,Chapter 5 in Comprehensive Rock
Engineering: Principles, Practice and Projects, Vol. II,Analysis and Design Method, ed. C. Fairhurst,
Pergamon Press, pp. 113-171, 1993
r/rw
Radialtensilestress,
MPa
0 2 4 6 8 10 12 14 16 18 20
Rate increase
3.5
3
2.5
2
1.5
1
0
0.5