1. INTRODUCTION With the rapid increase in oil and gas consumption, it has become necessary to drill deeper and in more challenging environments with high temperature and pressure. Drilling problems (stuck of pipe, lost circulation et al.) associated with the deep wells through shale formation are more difficult to control. In addition to insufficient mud weight, well instability is influenced by the temperature and solute concentration of the drilling mud [10, 11, 12]. Shales act as a semi-permeable membrane and the chemical osmotic effects can cause the fluid flow into or out the shale formation depending on the chemical potential of the drilling mud and formation [3, 9, 12, 14, 19]. Swelling pressure contributes to the shale instability while drilling [22]; swelling pressure can be changed by the complex physical and chemical interactions when the shale is exposed to the drilling fluid [11, 15, 22]. The thermal expansion/contraction of shale and fluid also contributes the wellbore instability [10, 12, 17]. Shales have very low permeability (order of nano-darcy), so convective fluid flow is not the dominant

fluid movement and coupled flows are also important. For low permeability shales with good membrane characteristics, diffusion is the main ion transfer style, and heat conduction is the dominant thermal energy transport. Thermal effects on the stress/pore pressure distribution and wellbore instability have been investigated using porothermoelastic models which neglect the convective fluid and heat transport [11, 17]. Chemical effects on the stress/pore pressure distribution and wellbore instability also have been investigated using chemoporoelastic models which combine chemical osmosis and ion diffusion in poroelasticity while neglecting the convective fluid flow and ion transfer [11, 15]. While drilling at high temperature and pressure, both thermal and chemical effects need to be considered in designing safe mud properties. Ghassemi et al. [10] developed a coupled Chemo-Poro-Thermoelastic model which can be used to analyze the combined of thermal and chemical effects on the stress/pore distribution around a wellbore. This paper applies the model to analyze wellbore instability and provides an improved approach to optimize the mud weight, temperature and salinity.

ARMA/USRMS 06-916

Optimization of Mud Properties for Drilling in Shale Using Coupled

Chemo-Poro-Thermoelasticity Tao, Q. and Ghassemi, A. Department of Geology & Geological Engineering, University of North Dakota, Grand Forks, ND 58202, USA

Copyright 2006, ARMA, American Rock Mechanics Association This paper was prepared for presentation at Golden Rocks 2006, The 41st U.S. Symposium on Rock Mechanics (USRMS): "50 Years of Rock Mechanics - Landmarks and Future Challenges.", held in Golden, Colorado, June 17-21, 2006. This paper was selected for presentation by a USRMS Program Committee following review of information contained in an abstract submitted earlier by the author(s). Contents of the paper, as presented, have not been reviewed by ARMA/USRMS and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of USRMS, ARMA, their officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMA is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented.

ABSTRACT: Borehole instability is a widespread problem in petroleum industry, especially when drilling in deep, low strength shale formations at high temperature and pressure. The instability can be caused by the high compressive effective stress or tensile stress due to the stress concentration and pore pressure increase while drilling. Mud temperature and salinity also directly and indirectly impact the distribution of stress and pore pressure around the wellbore. In this work, a Chemo-Poro-Thermoelastic stress analysis is implemented into a wellbore stability model to quantitatively optimize mud weight, salinity, and temperature. The results show that drilling with a cooler and more saline mud enhances wellbore stability, and the range of safe mud weight window; whereas drilling with warmer and lower saline mud tends to increase the rock failure and wellbore instability. Manipulation of both temperature and chemistry make it possible to maintain a wellbore stable by cooling a lower salinity mud or increasing the salinity of a warmer mud.

2. A COUPLED CHEMO-PORO-THERMOELASTIC MODEL

The shale-fluid system is simplified, and the shale matrix is assumed to be saturated with a binary electrolyte fluid consisting of a solute and diluent, and to have semi-permeable membrane behavior with swelling characteristic. All forms of convective transport are neglected, and three direct fluxes are considered: (1) heat flow driven by the temperature gradient; (2) solute flux driven by the solute concentration gradient, and temperature gradient (thermal filtration); (3) fluid flow driven by the pore pressure gradient and the solute concentration gradient.

Heat flow has a direct effect on the stress and strain change in the shale matrix, and the volumetric variation in the fluid and the pore space, therefore, it engenders fluid flow. Solute flux also results in fluid flow by directly causing pore pressure changes. The coupled constitutive equations of the non-isothermal shale-fluid system based on the first and second laws of thermodynamics have been derived by Diek and Ghassemi [8]. In the linear version, the responses of stress σij and pore volume ζ to the strain components εij, pore pressure p, solute mass fraction Cs, and temperature T are given by:

ijS

ijijkkijij TCpGKG δγχδαδεεσ &&&&&& 1')

32(2 −+−−+=

(1)

TCp S &&&&&2

'' γχβεαζ −++= (2)

The coefficients in the constitutive Equations (1) and (2) are:

ρ

ω−α=α

fD

S

RTC

M__

0

__0'

0

____0' )1(

TCKR

MK

Q

fD

S

f ρ

−αω+

φ+=β

)1( __

__

__0

D

S

S C

C

C−ω=χ

χαχK

1' −=

0

__0

1

RTC

MsKD

So

mωαγ +=

0

__0

2)1()(

KRTC

MsD

So

mfm−+−+= αωφααααγ (3)

Where ___

SC and ____

DC denote the mean values of the solute and diluent mass fraction, respectively. os is the reference value of the specific fluid entropy at the average system temperature and solute mass fraction, T0 is the average absolute temperature of the system. According to the coupled constitutive equations, any of the three fluxes can cause changes in: (i) stress/strain in the shale matrix, (ii) pore volume and the pore pressure; and the stress/strain changes also induce volume change in the pore space, causing pore pressure change and fluid flow. So, the model can be used to analyze a complex coupled hydraulic-mechanic-thermal-chemical system.

3. STRESS AND PORE PRESSURE SOLUTIONS AROUND A WELLBORE

For a borehole drilled in shale, the solutions of stress and pore pressure around the wellbore are obtained analytically [10] using a procedure similar to poroelasticity [2, 7] by combining and solving the constitutive and transport equations of the shale-fluid system subject to the appropriate boundary and initial conditions (Eq. (3)). For the current problem the BC�s are:

=∆=∆=∆

=∆=∆=∆

∞→

−=∆−=∆

−=∆−=∆

−=∆

−=∆=

∞

∞

∞

000

000

TCp

rat

TTTCCC

PPp

Parat

r

rz

rr

shm

shmshm

rr

rzrz

rrmrr

θθθ σσσ

σσσσ

σσ (3)

The analytical solutions are in the Laplacian domain and the numerical results in the time domain for real examples can be obtained by applying Stehfest inversion method [4].

4. WELLBORE INSTABILITY ANALYSIS USING THE MODEL

The temporal stress and pore pressure distribution around any wellbore under thermal and chemical loading can be obtained by applying the appropriate physical and chemical parameters and loadings, and inverting the solutions into time domain, and then used to analyze the possible failure which is determined by the stress state and the strength of the rock. If we vary the mud properties (mud weight, temperature and salinity) and judge whether the wellbore is stable or not, the safe range of mud properties can be obtained. A complete treatment of the stability analysis and mud optimization for poroelastic and water sensitive shale has been presented by Ghassemi et al. [13]. 4.1 Rock failure Wellbore instability includes compressive failure, hydraulic fracturing (tensile fracture along the radial direction) and radial spalling (tensile fracture perpendicular to the radial direction). Compressive failure and hydraulic fracturing cause the most common drilling problems namely, stuck pipe (collapse of the wellbore) and lost circulation, respectively. In reality, the tensile fracture inside the formation may not be exactly along the radial direction or perpendicular to it, but it is important to distinguish between the two because they result in different drilling problems � hydraulic fracturing causes lost circulation and radial spalling causes local failure of the formation and possible enlargement of the wellbore. In this paper the Drucker-Prager failure criteria [1] is used to determine whether the compressive failure occurs, and a compressive shear failure potential (SFP) is also defined (Eq. (4)): bImJSFP d −−= 12 (4) Where

'''1 zzrrI σσσ θθ ++=

[ ]222

2''2''2''2 )()()(

61

zrzr

zzzzrrrrJ

θθ

θθθθ

σσσ

σσσσσσ

+++

−+−+−= (5)

md and b are coefficients that depend on the cohesion c and the angle of internal friction, φ . SFP is the compressive failure potential,

compressive shear failure occurs when 0≥SFP (all results presented in the paper use the geomechanics sign convention, i.e., compression positive), and the maximum SFP around the wellbore (r/a=1.0~ 1.5) is taken as the SFP of the wellbore. The radial spalling occurs when the radial effective stress is less than 0, and the radial spalling potential is defined by the radial effective stress ( '

rrRSP σ= ). In the same way the hydraulic fracturing potential (HFP) is defined in terms of the tangential stress ( '

θθσ=HFP ). The minimum RSP and HFP around the wellbore (r/a=1.0~ 1.5) are taken as the RSP and HFP of the wellbore. 4.2. Thermal and chemical effects on rock failures around a wellbore A section of a vertical wellbore at a depth of 1 km is considered and failure potentials (SFP, RSP and HFP) are calculated to assess the wellbore stability under different hydraulic, thermal and chemical conditions (the in-situ stress and the related rock and fluid parameters are listed in Table 1). Experimental data on the full range of Chemo-Poro-Thermoelastic properties of shale are not commonly available. The set used here has been extracted from Cui et al. [5, 6]; van Oort et al. [23], Tanner [21] and Lide [18]. 4.2.1. Porothermoelastic effects Porothermoelastic theory [16, 17, 20] was developed by incorporating the heat transport into Biot�s poroelastic theory and coupling the thermal expansion/contraction of shale matrix and pore fluid with the hydraulic-mechanic processes. Thermal expansion tends to increase the tangential stress around the wellbore whereas thermal contraction decreases it, therefore, the variation of mud temperature can impact the rock failure and the wellbore stability. Thermal expansion/contraction also changes the pore pressure which can affect the wellbore stability by changing the effective stresses. Figure 1 shows the failure potential as a function of mud pressure at 1 hour after drilling. The critically low mud pressure causing compressive failure and radial spalling, and the critically high mud pressure causing hydraulic fracturing can be obtained from the figure to decide on the safe drilling mud pressure (Table 2).

Table 1. Properties of shale formation and drilling mud.

In-situ stresses (σv, σH, σh ) 25 MPa/km, 29MPa/km, 20MPa/km

Pore pressure 10 MPa/km Shear modulus G 760 MPa Biot�s coefficient α 0.966 Drained Poisson�s ratio ν 0.219 Undrained Poisson�s Ratio νu 0.461 Permeability coefficient κ 0.333 x10-17 m2/Pa.s Porosity φ 0.2989 Mean solute fraction CS 0.15

Fluid mass density __

fρ 1111.11 kg/m3

Fluid bulk modulus Kf 3291 MPa Reflection coefficient ℜ 0.2 Molar mass of solute (NaCl) Ms

0.0585 kg/mole

Swelling coefficient ω0 1.5 MPa Solute diffusivity DS 2.0x10-9 m2/s Thermal expansion coefficient of solid mα

1.8x10-5 K-1

Thermal expansion coefficient of fluid fα

3.0x10-4 K-1

Thermal diffusivity cT 1.6x10-6 m2/s

Coefficient of thermal diffusion DT

6.0x10-12 m2/(s.K)

Specific entropy (NaCl, CS =0.15) S0

3686 J/(kg.K)

Skempton�s coefficient B 0.915 Fluid diffusivity cf 6.0 x10-9 m2/s Fluid bulk modulus Kf 3290 MPa Elastic modulus E 1853 MPa Bulk modulus K 1099 MPa Solid bulk modulus Ks 32600 MPa Drucker-Prager material constant md

0.14

Drucker-Prager material constant b

12.0 MPa

-10

-5

0

5

10

15

0 5 10 15 20 25 30

Porothermoelastic, Tm=115 °C, Tsh=65 °CPoroelasticPorothermoelastic, Tm=65 °C, Tsh=115 °CElastic

Time = 1 Hr

Mud pressure (MPa)

Shea

r fai

lure

pot

entia

l (M

Pa)

Thermal effects on shear failure

1.a

-10

0

10

20

30

0 5 10 15 20 25 30

Porothermoelastic, Tm=115 °C, Tsh=65 °CPorothermoelastic, Tm=65 °C, Tsh=115 °CPoroelasticElastic

Time = 1 Hr

Mud pressure (MPa)

Hyd

raul

ic fr

actu

ring

pote

ntia

l (M

Pa)

Thermal effects on hydraulic fracturing

1.b

-20

-10

0

10

20

0 5 10 15 20 25 30

Porothermoelastic, Tm=115 °C, Tsh=65 °CPorothermoelastic, Tm=65 °C, Tsh=115 °CPoroelasticElastic

Time = 1 Hr

Mud pressure (MPa)

Rad

ial s

pallin

g po

tent

ial (

MPa

)

Thermal effects on radial spalling

1.c

Fig. 1. Thermal effects on the rock failures.

The critical values are those at which the potential intersects the y=0 line. Heating increases the maximum effective principal stress in the tangential direction and reduces the minimum effective principal stress in the radial direction (the latter could become tensile at short time), and so tends to cause compressive failure and radial spalling, increasing the critical low mud pressure required to prevent rock failure. Cooling reduces the effective tangential stresses in tangential direction, therefore enhances the hydraulic fracturing, but increases the effective radial stress, thereby inhibits the radial spalling. 4.2.2. Chemoporoelastic effects In addition to the poroelastic process, there are complex physico-chemical interactions between the shale and the fluid in the system, which can change the swelling pressure and cause the swelling or

shrinkage of the shale matrix depending on the change of the ion concentration in the pores [22]. The physico-chemical interactions can alter the effective stress by changing swelling pressure, thereby impacting on the stability of the wellbore. Figure 2 shows that drilling with higher salinity reduces the lower bound (critical low) mud pressure that causes compressive failure and radial spalling and increases the upper bound (critical high) mud pressure causing the hydraulic fracturing, therefore enhances the safe drilling range of mud pressure (Table 2). It is interesting to note that (Figure 2.c) so there is no safe drilling mud pressure with respect to radial spalling when drilling with a lower salinity mud (Table 2). 4.2.3. Coupled thermal and chemical effects Chemical swelling depends not only on the gradient of ion concentration (chemoporoelaticity), but also on the gradient of temperature [10]. Heat transport also has a direct effect on the ion concentration, and thermal filtration can cause an ion flux [21]. So, the above two models are unable to analyze rock failure and wellbore stability under both thermal and chemical loadings. However the coupled Chemo-Poro-Thermoelastic model used herein considers the couplings between the thermal and chemical processes, and is a robust tool for assessing rock failure and wellbore stability analysis in complicated thermal and chemical environments.

-10

-5

0

5

10

15

0 5 10 15 20 25 30

Chemoporoelastic, Cm=0.2, Csh=0.1Chemoporoelastic, Cm=0.1, Csh=0.2PoroelasticElastic

Time = 1 Hr

ω0=1.5 MPa

ℜ = 0.2

Mud pressure (MPa)

She

ar fa

ilure

pot

entia

l (M

Pa)

Chemical effects on shear failure

2.a.

-10

0

10

20

30

0 5 10 15 20 25 30

Chemoporoelastic, Cm=0.2, Csh=0.1Chemoporoelastic, Cm=0.1, Csh=0.2PoroelasticElastic

Time = 1 Hr

ω0=1.5 MPa

ℜ = 0.2

Mud pressure (MPa)

Hyd

raul

ic fr

actu

ring

pote

ntia

l (M

Pa)

Chemical effects on hydraulic fracturing

2.b.

-20

-10

0

10

20

0 5 10 15 20 25 30

Chemoporoelastic, Cm=0.2, Csh=0.1Chemoporoelastic, Cm=0.1, Csh=0.2PoroelasticElastic

Time = 1 Hr

ω0=1.5 MPa

ℜ = 0.2

Mud pressure (MPa)

Rad

ial s

palli

ng p

oten

tial (

MPa

)

Chemical effects on radial spalling

2.c. Fig. 2. Chemical effects on rock failure.

Figure 3 shows that drilling with a cooler low salinity mud decreases the lower bound (critical low) mud pressure and raises the upper bound (critical high) mud pressure, resulting in a wilder range for the safe drilling mud pressure. A porothermoelastic model predicts that cooling tends to increase tensile failure and the upper bound (critical high) mud pressure, but a Chemo-Poro-Thermoelastic model indicates the opposite (Figure 3.b), because a cooler mud reduces the swelling pressure (by increasing chemical potential according to the Gibbs-Duhem equation for non-isothermal condition) and increases the effective stress, thereby preventing the tensile failure. Figure 4 shows the failure potential as a function of mud salinity when the mud pressure and temperature are kept constant. When the mud pressure is maintained as 10 MPa, and the

temperature of mud and shale are 85 °C and 95 °C respectively, the shear failure potential decreases and becomes negative with increasing mud salinity. Meanwhile, radial spalling potential increases from negative to positive with the increase in the mud salinity; so both shear failure and radial spalling might be prevented by increasing the mud salinity (note that shear failure occurs when shear failure potential is greater than zero, but tensile failure occurs when tensile failure potential is less than zero). The hydraulic fracture increases with the increasing of mud salinity, so increasing mud salinity may prevent hydraulic fracturing. 4.3. Time-dependent effects The diffusive processes of heat transport, ion transfer, and fluid flow are time-dependent, and any disturbance of stress, pore pressure, temperature, or ion concentration will result in time-dependent fluxes, and stress/pore pressure distribution. So the rock failure determined by the local effective stresses is also time-dependent. Figure 5 shows that the failure potential for the case drilling with cooler and more saline mud are time dependent, the upper bound of mud pressure for hydraulic fracturing decreases with time, and the lower bound for compressive failure first decreases (from 1 hr to 24 hrs), and then increases (from 24 hrs to 120 hrs). It can also bee seen that there is no potential for radial spalling at large time.

-10

-5

0

5

10

15

0 5 10 15 20 25 30

Chemo-Poro-Thermoelastic, Tm=115 °C, Tsh=65 °CChemo-Poro-Thermoelastic, Tm=65 °C, Tsh=115 °CChemoporoelastic, Cm=0.1, Csh=0.2

Dashed: Cm=0.1, Csh=0.2

Solid: Cm=0.2, Csh=0.1

Time = 1 Hr

ω0=1.5 MPa

ℜ = 0.2

Mud pressure (MPa)

She

ar fa

ilure

pot

entia

l (M

Pa)

Coupled thermal and chemical effects on shear failure

3.a.

-10

0

10

20

30

0 5 10 15 20 25 30

Chemo-Poro-Thermoelastic, Tm=115 °C, Tsh=65 °CChemo-Poro-Thermoelastic, Tm=65 °C, Tsh=115 °CChemoporoelastic, Cm=0.1, Csh=0.2

Dashed: Cm=0.1, Csh=0.2

Solid: Cm=0.2, Csh=0.1

Time = 1 Hr

ω0=1.5 MPa

ℜ = 0.2

Mud pressure (MPa)

Hyd

raul

ic fr

actu

ring

pote

ntia

l (M

Pa)

Coupled thermal and chemical effects on hydraulic fracturing

3.b.

-30

-20

-10

0

10

0 10 20 30

Chemo-Poro-Thermoelastic, Tm=65 °C, Tsh=115 °CChemo-Poro-Thermoelastic, Tm=65 °C, Tsh=115 °CChemoporoelastic

Dashed: Cm=0.1, Csh=0.2

solid: Cm=0.2, Csh=0.1

Time = 1 Hr

ω0=1.5 MPa

ℜ = 0.2

Mud pressure (MPa)

Rad

ial s

palli

ng p

oten

tial (

MP

a)

Coupled thermal and chemical effects on radial spalling

3.c.

Fig. 3. coupled thermal and chemical effects on rock failure.

4.4. Temperature and salinity dependent mud weight window The mud window is the mud weight between the lower and upper bound values and represents the mud weights that theoretically prevent wellbore instabilities. Predicting the right mud weight window is critical to drill successfully, however the thermal and chemical impacts on the rock failure make the prediction even more difficult. Rock failure is not only dependent on the mud weight, but also on the mud temperature and salinity, so the thermal and chemical effects can not be neglected when constructing the mud weight window.

In this work, the mud weight windows are constructed by varying the inclination angle of the wellbore from 0° to 90° (0° is for vertical wellbore and 90° is for horizontal wellbore), in the σH � σv plane. All of relevant parameters are listed in Table 1. Figure 6 and Figure 7 show that the width of the mud weight window decreases with the increasing temperature when Cm=0.2, Csh=0.1, and increases with increasing mud salinity when Tm=85, Tsh=95. So the wellbore could be maintained stable by increasing the mud salinity or reducing the mud temperature.

-10

0

10

20

0 0.1 0.2 0.3 0.4 0.5

Hydraulic fracturing potentialRadial spalling potentialShear failure potential

Dashed: ω0=1.5 MP

Solid: ω0=8.0 MP

Time = 1 Hr

ℜ = 0.2

Mud pressure = 10 MPa

Tm=85 °C, Tsh=95 °C

Csh=0.15

Solute mass fraction of mud (Cm)

Failu

re p

oten

tial (

MP

a)

Faliure potential vs Mud salinity

Fig. 4. The failure potential as a function of mud salinity when mud weight is 10 MPa and Tm=85 °C, Tsh=95 °C.

-10

0

10

20

30

0 5 10 15 20 25 30

120 Hrs24 Hrs1 Hr

ω0=1.5 MPa

ℜ = 0.2

Cm=0.2, Csh=0.1

Tm=65 °C, Tsh=115 °C

Circle: Radial spalling potential

Triangle: Shear failure potential

Rectangle: Hydraulic fracturing potential

Mud pressure (MPa)

Failu

re p

oten

tial (

MP

a)

Temporal failure potential vs Mud pressure

Fig. 5. Time-dependent rock failure.

0.5

1.0

1.5

2.0

2.5

0 20 40 60 80 100

Tm=94 °C, Tsh=86 °CTm=93 °C, Tsh=87 °CTm=91 °C, Tsh=89 °CTm=90 °C, Tsh=90 °CTm=85 °C, Tsh=95 °C

Time = 1 Hr

ω0=1.5 MPa

ℜ = 0.2

Dashed: Critical low mud weight

Solid: Critical high mud weight

Cm=0.2, Csh=0.1

Inclination (° )

ρ mud

/ ρ w

ater

Mud weight window

Fig. 6. mud weight window for varied mud temperature.

0

0.5

1.0

1.5

2.0

2.5

0 20 40 60 80 100

Cm=0.2, Csh=0.1Cm=0.1, Csh=0.2Cm=0.065, Csh=0.235Cm=0.06, Csh=0.24Cm=0.055, Csh=0.245Cm=0.05, Csh=0.25

Time = 1 Hr w0=1.5 MPa ℜ Â= 0.2

Dashed: Critical low mud weight

Solid: Critical high mud weight

Tm=85 °C, Tsh=95 °C

Inclination (° )

ρ mud

/ ρ w

ater

Mud weight window

Fig. 7. mud weight window for varied mud salinity.

5. DISCUSSIONS AND CONCLUSIONS A coupled theory that considers thermal expansion/contraction, chemical osmosis, and physico-chemical interactions between the mud and shale has been used to assess wellbore stability while drilling at high temperature and high pressure environment. The analysis includes optimization of the mud temperature, salinity and weight. Using Drucker-Prager compressive failure criterion and tensile failure criterion, the impacts of mud temperature, salinity and pressure on the rock failure have been analyzed. The results suggest that cooling tends to prevent compressive failure, radial spalling and hydraulic fracturing, whereas heating tends to enhance them. Also, drilling with a higher salinity mud reduces the swelling pressure, thereby enhancing the wellbore stability, and drilling with lower salinity reduces the wellbore stability. Furthermore, the interaction between thermal and chemical phenomena can be used to maintain a

wellbore stable while drilling; lowering salinity when the mud is cooler than the formation and increasing salinity if the mud is warmer. The analytical nature of solutions used in the model facilitates real-time wellbore stability assessment.

REFERENCES 1. Boresi, A.P. and R.J. Schmidt, 2003. Advanced

Mechanics of Materials. 6th ed. John wiley & sons, INC.

2. Carter J.P. and J.R. Booker, 1982. Elastic consolidation around a deep circular tunnel. International Journal of Solids and Structures, 18(12), 1059-1074.

3. Chenevert, M.E., 1970. Shale control with balanced-activity oil-continuous muds. J. Pet. Technol., 1309-1316.

4. Cheng, A.H.-D., Sidauruk, P., and Abousleiman, Y., 1994. Approximate inversion of the Laplace transform. Mathematica Journal, Vol. 4, No. 2, 76-82.

5. Cui, L., A.H.-D. Cheng, and Y. Abousleiman, 1997. Poroelastic solutions for an inclined borehole. J. Appl. Mech. 64, 32�38 (transactions of the ASME)

6. Cui, L., Ekbote, S., Abousleiman, Y., Yaman, M., and Roegiers, J.-C., 1998. Borehole stability analyses in fluid saturated formations with impermeable walls. Int. J. Rock Mech. Min. Sci. 35, 4�5, Paper No. 061 .

7. Detournay, E. and A.H.-D. Cheng , 1988. Poroelastic response of a borehole in a non-hydrostatic stress field. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 25, 171�182.

8. Diek, A., and A. Ghassemi (2004). A thermoelastic theory of fluid saturated argillaceous rock. Dept. of Geology & Geological Engineering, University of

North Dakota, Internal Report NDRM-WSR-04-01, 38 pp.

9. Fritz, S.J., and I.W. Marine. 1983. Experimental support for a predictive osmotic model of clay membranes. Geochim. Cosmochim. Acta 47:1515�1522.

10. Ghassemi, A., Q., Tao and A. Diek, 2005. Stress and Pore Pressure Distributions around a Wellbore in Shale due to Coupled Chemo-Poro-thermoelastic Processes (Submitted to J. Pet. Sci. Engrg.).

11. Ghassemi, A. and A. Diek, 2003. Linear chemo-poroelasticity for swelling shales: theory and application. J. Pet. Sci. Eng. 38, 123�135.

12. Ghassemi, A. and A. Diek, 2002. Poro-thermoelasticity for swelling shales. J. Pet. Sci. Eng. 34, 123�135.

13. Ghassemi, A., A. Diek, A.C. Wolfe, and J.-C. Roegiers, 1999. A chemo-mechanical model for borehole stability analyses. In: Vail Rocks' 99, 37th U.S. Rock Mechanics Symposium, Balkema, The Netherlands, 239�246.

14. Hanshaw, B.B. and E. Zen, 1965. Osmotic equilibrium and overthrust faulting. Geol. Soc. Amer. Bull. 76, 1379�1387.

15. Heidug, W.K. and S.-W. Wong, 1996. Hydration swelling of water-absorbing rocks: a constitutive model. Int. J. Numer. Anal. Methods Geomech. 20, 403�430.

16. Kurashige, M., 1989. A thermoelastic theory of fluid-filled porous materials. Int. J. Solids Struct. 25 (9), 1039�1052.

17. Li, X., L. Cui and J.-C. Roegiers, 1998. Thermoporoelastic modeling of wellbore stability in non-hydrostatic stress field. Int. J. Rock Mech. Min. Sci. 35, 4�5, Paper No. 063 .

Table 2. critical low and high mud pressures. Mud pressure (MPa) Material model Loading condition

Shear failure

Hydrauclic fracturing

Radial spalling

Safe drilling

elastic N/A ≤12.5 ≥21.0 ≤10.0 12.5~21.0 poroelastic N/A ≤11.3 ≥22.4 ≤14.0 14.0~22.4

Tm=65 °C, Tsh=115 °C ≤11.0 ≥21.9 ≤5.0 11.0~21.9 porothermoelastic Tm=115 °C, Tsh=65 °C ≤14.1 ≥23.0 ≤22.0 22.0~23.0

Cm=0.1, Csh=0.2 ≤12.3 ≥22.3 Any N/A chemoporoelastic Cm=0.2, Csh=0.1 ≤10.2 ≥22.7 ≤11.1 11.1~22.7

Tm=65 °C, Tsh=115 °C Cm=0.1, Csh=0.2

≤4.7 ≥23.1 ≤5.2 5.2~23.1

Tm=65 °C, Tsh=115 °C Cm=0.2, Csh=0.1

≤2.7 ≥23.5 ≤1.2 2.7~23.5

Tm=115 °C, Tsh=65 °C Cm=0.1, Csh=0.2

≤21.8 ≥21.5 Any N/A

Chemo-Poro-Thermoelastic

Tm=115 °C, Tsh=65 °C Cm=0.2, Csh=0.1

≤19.7 ≥21.8 Any N/A

18. Lide, D.R., Editor in Chief, 2000-2001. CRC Handbook of Chemistry and Physics. 81st Edition. CRC Press, Inc.

19. Lomba, R.F.T., M.E. Chenevert and M. Sharma, 2000. The role of osmotic effects in fluid flow through shales. J. Pet. Sci. Eng. 25, 25�35.

20. McTigue, D., 1986. Thermoelastic response of fluid-saturated porous rock. J. Geophys. Res. 91 B9, 9533�9542.

21. Tanner, C.C., 1953. The soret effect. Trans. Faraday Soc. v. 49, 611-619.

22. Van Oort, E., 2003. On the physical and chemical stability of shales. J. Pet. Sci. Eng. 38, 213�235.

23. van Oort, E., A.H. Hale, and F.K. Mody, 1994. Critical parameters in modeling the chemical aspects of borehole stability in shales and in designing improved water-based shale drilling fluids. In: SPE Paper # 28309, Richardson, TX .