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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

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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.

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Winters, T.-S. Yun, Physical properties of

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10. L.I. Rubinstein, The Stefan Problem, AmericanMathematical Society,1971

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12. A. Ghassemi, Q. Tao, A. Diek, Influence of coupledchemo-poro-thermoelastic processes on pore pressureand stress distributions around a wellbore in swelling

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13. Qingfeng Tao, Wellbore Stability in water sensitiveshales, Master of Science Thesis, University of North

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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