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[a] Key Laboratory of Education Ministry for Enhanced Oil Recovery， Northeast Petroleum University， Daqing， Hei Longjiang Province， China.
*Corresponding author.
Supported by National Natural Science Foundation of China， “The study on the fractal evolution of pore-fracture and properties of porosity-permeability under hydraulic fracturing in coal bed gas wells basing on the chaos theory （51274067）”.
Received 2 July 2013； accepted 23 August 2013
Abstract
Considering that drilling fluid filtration and the temperature difference between borehole wall rock and drilling fluid can cause the stress variation of the borehole wall. The stress distribution model was derived under the effect of thermal-flow-solid coupling. The safe mud density window calculating model considering pore pressure and temperature difference variation was established according to Moore-Coulomb criterion and borehole wall rock tensile failure criterion. The result calculated by the model can be expressed as follow. （1） When the temperature difference between borehole rock and drilling fluid is constant， with the enhancement of fluid filtration， borehole rock pore pressure increasing， the collapse pressure increasing， breakdown pressure decreasing， the stability of the borehole becomes deteriorating. （2） When the borehole wall rock pore pressure is constant， if drilling fluid makes wall rock temperature decreasing， with the temperature difference increasing， both the collapse pressure and breakdown pressure decreasing， the stability of the borehole becoming deteriorating， it is not conducive to drilling safely. If drilling fluid make wall rock temperature increasing， with the temperature difference increasing， both the collapse pressure and breakdown pressure increasing， the borehole tending to stabilize， it is conducive to drilling safely.
Key words： Pore pressure； Temperature difference； Borehole stability； Safe mud density window
Ai， C.， Li， Y. W.， & Liu， Y. （2013）. The Effects of Pore Pressure and Temperature Difference Variation on Borehole Stability. Advances in Petroleum Exploration and Development， 6（1），-0. Available from： URL： http：//www.cscanada.net/index.php/aped/article/view/j.aped.1925543820130601.1546
DOI： http：//dx.doi.org/10.3968/j.aped.1925543820130601.1546
INTRODUCTION
The study of borehole stability is a hotspot and difficult problem in drilling engineering[1-6]. In recent years， with the global economic growth， the use of oil and gas resources is increasing. Oil and gas exploration and development focus is gradually shifting to the complex geological conditions and unconventional reservoirs. The drilling environment is becoming more and more complex. However， in the process of drilling， under the effects of the drilling fluid in the borehole， the pore pressure and temperature can change， which causes stress redistribution of the borehole， and the stability of the borehole can be affected， it is easily causing well blowing， mud loss， borehole collapse and sticking accidents. Scholars at home and abroad have carried out a more comprehensive study on borehole stability. The borehole stability mechanism has been studied from fluid-solid coupling， thermal-flow-solid coupling and mechanics-chemical coupling. Xinglong Wang and Yuanfang Cheng[7] have established borehole wall temperature and pressure coupling mechanical model in shale formations， illustrating that temperature changes can cause the variation of pore pressure， changing the borehole wall stress distribution by numerical method. Baohua Wei[8] has analyzed the effect of temperature and pressure on borehole stability， establishing additional stress calculation model caused by pore pressure changes. Lewen Zhang[9] has established mechanics-chemical coupling mathematical model for calculating borehole wall stress， the wellbore wall pore pressure and stress variation regulations have been analyzed under the effect of mechanics-chemical coupling. Although some scholars have discussed the effect of pore pressure and temperature difference variation on borehole stability， the theory of effects of pore pressure and temperature changes on porosity and wall stress is limited， and the borehole stability model considering simultaneous changes of pore pressure and temperature has not been reported. In this paper， the problem has been better explained in theory， which is a supplement and completeness for the existing borehole stability mechanism. 1. THE MECHANICAL MODEL OF BOREHOLE ROCK
In the process of drilling， the mechanical model of borehole rock is shown in Figure 1. Assuming that formation rock is isotropic elastic medium， the horizontal maximum principal stressand minimum principal stressare loaded on formation rock. pi is the pressure caused by drilling fluid column in borehole， pp is the initial formation pore pressure.
In the process of drilling， the key factor affects borehole stability is the variation of formation temperature and pressure. On the one hand， in order to prevent well blowing， mud loss， it is required that the drilling fluid column pressure is greater than the pore pressure， which can lead to drilling fluid filtrate seep into formation， and it can generate percolation zone around the borehole （gray area within the dotted line in Figure 1）， and the solid phase particles in drilling fluid are left on the borehole wall in the forms of mud cake （the black ring region in Figure 1）. On the other hand， the temperature difference between the deep formation and fluid can generate additional stress and strain on borehole wall rock on the basis of the original balance， which can change the original bearing force balance.
2. THE BOREHOLE WALL STRESS DISTRIBUTION CONSIDERING PORE PRESSURE AND TEMPERATURE DIFFERENCE VARIATION
Pore pressure and temperature difference variation will cause the wall stress distribution variation. Considering in situ stress， drilling fluid column pressure， fluid seepage and thermal stress effects separately that can cause borehole wall stress variation， borehole wall stress is analyzed.
2.1 Borehole Wall Stress Distribution Caused by in Situ Stress
When the borehole is opened， the stress concentration can generate around the borehole wall under in situ stress field， wall rock stress state can be expressed as follow[10].
Where： σr is radial stress of the wall rock， MPa； σθ is circumferential stress of the wall rock， MPa； σz is the vertical stress of the wall rock， MPa； σH is the horizontal maximum principal stress， MPa； σh is the horizontal minimum principal stress， MPa； σv is overburden stress， MPa； r is the distance between any point on the borehole wall rock and the center of the borehole， m； R is the radius of the borehole， m； θ is the included angle between any point on the borehole wall rock and the horizontal maximum principal stress， （°）； is the poisson ratio of the rock.
2.2 Borehole Wall Stress Caused by Drilling Fluid Column Pressure and Seepage Effects Under the effects of drilling fluid column pressure and the additional stress caused by fluid seepage， the stress distribution caused by wall surrounding area can be expressed as follow[10].
where： Pi is drilling fluid column pressure in borehole， MPa； Pp is formation initial pore pressure， MPa； α is effective stress factor； is the porosity of the wall rock.
2.3 The Thermal Stress Analysis of the Borehole Wall
The borehole wall stress variation caused by the temperature difference between drilling fluid and borehole rock can be expressed as follow.
Where： E is Young’s modulus of borehole rock， MPa； T is the temperature of borehole rock that has change， （℃）； T0 is the initial temperature of borehole rock， （℃）.
2.4 The Porosity Variation of Borehole Rock
In the process of drilling， the porosity of rock will change under the effect of pore pressure and thermal stress. By changing the porosity that has changed can be expressed as follow.
Where： is the porosity of the rock that has changed； is the initial porosity of the rock； is the pore pressure variation difference caused by drilling fluid filtration， MPa.
2.5 The Stress Distribution of Borehole Wall
The rock stress distribution can be obtained by the superposition of the various parts of the stress according to linear superposition theory in elastic mechanics theory.
The effects of pore pressure and temperature difference of the rock on borehole rock stress can be obtained by analyzing the above equation， then study their effects on safety mud density window by analyzing stress variation of the borehole wall.
3. THE DETERMINATION OF SAFE MUD DENSITY WINDOW
The different stress states can be obtained by borehole wall stress distribution model. The caving pressure can be obtained by deriving Moore-Coulomb criterion. The breakdown pressure of the borehole wall can be gotten according to the tensile failure criterion. The safety mud density window calculating model is established considering fluid seepage and temperature variation on the condition that the overburden stress is the intermediate principal stress.
By deriving the drilling fluid density that is equivalent to the caving pressure can be expressed as follow.
Where： ρm is the drilling fluid density that is equivalent to the caving pressure， g/cm3；， is the angle of internal friction of the rock， （°）； is the well depth， m； C0 is cohesive force of the rock， MPa. The drilling fluid density that is equivalent to the breakdown pressure can be expressed as follow.
Where： Pf is the drilling fluid density that is equivalent to the breakdown pressure， g/cm3； is the tensile strength of the rock， MPa.
4. THE EFFECTS OF PORE PRESSURE AND TEMPERATURE DIFFERENCE VARIATION ON BOREHOLE STABILITY
In order to analyze the effects of pore pressure and temperature difference variation on borehole stability， the rock mechanics parameters are obtained from the rock at the depth of 2400 m. The horizontal maximum principal stress is 48 MPa. The horizontal minimum principal stress is 38 MPa. Poisson’s ratio is 0.26. The Young’s modulus is 13700 MPa. The porosity is 0.2. The tensile strength of rock is 3.55 MPa. The cohesive force is 6.33 MPa. The angle of internal friction is 33.11°. The effective stress coefficient is 0.7. The thermal expansion coefficient is 0.00005. The initial formation pore pressure is 23.86 MPa.
The effects of borehole rock pore pressure and temperature difference variation on safe mud density window can be shown in Figure 2 and Figure 3. The result can be shown as follow： （1） When the temperature difference is constant， with the enhancement of fluid filtration， borehole rock pore pressure increasing， causing the collapse pressure increasing， breakdown pressure decreasing， the safe drilling fluid density window becoming smaller， and it is not conducive to drilling safely. Thus， for a narrow safe density window drilling， the building capacity of the fluid should be strengthened， reducing effects of fluid filtration on wellbore stability. （2） When the borehole wall rock pore pressure is constant， if the wall rock temperature increases， with the temperature difference increasing， both the collapse pressure and breakdown pressure increasing， safe drilling fluid density window becoming larger， it is conducive to drilling safely. If wall rock temperature is lowered， with the temperature difference increasing， both the collapse pressure and breakdown pressure decreasing， the safety drilling fluid density window becoming smaller， it is not conducive to drilling safely. It can be shown that the cooling effect of drilling fluid to the formation is not conducive to borehole stability.
CONCLUSION
When the temperature difference between borehole rock and drilling fluid is constant， with the enhancement of fluid filtration， borehole rock pore pressure increasing， causing the collapse pressure increasing， breakdown pressure decreasing， the safe drilling fluid density window becoming smaller， borehole stability becoming deterioration， it is not conducive to drilling safely. When the borehole wall rock pore pressure is constant， if drilling fluid makes wall rock temperature decreasing， with the temperature difference increasing， both the collapse pressure and breakdown pressure decreasing， the safety drilling fluid density window becoming smaller， borehole stability becoming deterioration， it is not conducive to drilling safely. If drilling fluid make wall rock temperature increasing， with the temperature difference increasing， both the collapse pressure and breakdown pressure increasing， safe drilling fluid density window becoming larger， the borehole wall tending to stabilize， it is conducive to drilling safely. REFERENCES
[1] Aadnow， B. S.， Rogaland， U.， & Chenevert， M.E. （1987）. Stability of Highly Inclined Boreholes. SPE 16052.
[2] Yew， C. H. （1989）. On Fracturing Design of a Deviated Wells. SPE 19722.
[3] Jia， G. Y.， Zhang， L. S.， & Duan， Y. X. （2006）. Drilling in High Density and Thin Pressure Windows Complex Formation. SPE 104411.
[4] Wang， Q.， Zhou， Y. C.， Wang， G.， et al. （2012）. The Flow-Solid-Thermal Coupling Model for Shale Borehole Stability. Petroleum Exploration and Development， 39（4）， 475-480.
[5] Li， Y. F.， Fu， Y. Q.， Tang， G.， et al. （2012）. The Regulations of Stress Type Affecting Borehole Stability for Directional Wells. Natural Gas Industry， 32（3）， 78-80.
[6] Li， H. F.， & Chen， M. （2011）. Real-Time Prediction of Borehole Instability Based on Actual Drilling Data. Acta Petrolei Sinica， 32（2）， 324-328.
[7] Wang， X. L.， Cheng， Y. F.， & Zhao， Y. Z. （2007）. The Effect of Temperature on Wellbore Stability in Shale During Drilling. Petroleum Drilling Techniques， 35（2）， 42-45.
[8] Wei， B. H.， Lu， X. F.， Wang， B. Y.， et al. （2004）. The Effect of Formation Temperature Variation on Wellbore Stability in High Temperature Wells. Drilling Fluid and Completion Fluid， 21（6）， 15-18.
[9] Zhang， L. W.， Qiu， D. H.， & Cheng， Y. F. （2009）. Research on the Wellbore Stability Model Coupled Mechanics and Chemistry. Journal of Shandong University （Engineering Science）， 39（3）， 111-114.
[10] Chen， M.， Jin， Y.， & Zhang， G. Q. （2008）. Petroleum Engineering Rock Mechanics （pp. 62-65）. Beijing： Science Press.
*Corresponding author.
Supported by National Natural Science Foundation of China， “The study on the fractal evolution of pore-fracture and properties of porosity-permeability under hydraulic fracturing in coal bed gas wells basing on the chaos theory （51274067）”.
Received 2 July 2013； accepted 23 August 2013
Abstract
Considering that drilling fluid filtration and the temperature difference between borehole wall rock and drilling fluid can cause the stress variation of the borehole wall. The stress distribution model was derived under the effect of thermal-flow-solid coupling. The safe mud density window calculating model considering pore pressure and temperature difference variation was established according to Moore-Coulomb criterion and borehole wall rock tensile failure criterion. The result calculated by the model can be expressed as follow. （1） When the temperature difference between borehole rock and drilling fluid is constant， with the enhancement of fluid filtration， borehole rock pore pressure increasing， the collapse pressure increasing， breakdown pressure decreasing， the stability of the borehole becomes deteriorating. （2） When the borehole wall rock pore pressure is constant， if drilling fluid makes wall rock temperature decreasing， with the temperature difference increasing， both the collapse pressure and breakdown pressure decreasing， the stability of the borehole becoming deteriorating， it is not conducive to drilling safely. If drilling fluid make wall rock temperature increasing， with the temperature difference increasing， both the collapse pressure and breakdown pressure increasing， the borehole tending to stabilize， it is conducive to drilling safely.
Key words： Pore pressure； Temperature difference； Borehole stability； Safe mud density window
Ai， C.， Li， Y. W.， & Liu， Y. （2013）. The Effects of Pore Pressure and Temperature Difference Variation on Borehole Stability. Advances in Petroleum Exploration and Development， 6（1），
DOI： http：//dx.doi.org/10.3968/j.aped.1925543820130601.1546
INTRODUCTION
The study of borehole stability is a hotspot and difficult problem in drilling engineering[1-6]. In recent years， with the global economic growth， the use of oil and gas resources is increasing. Oil and gas exploration and development focus is gradually shifting to the complex geological conditions and unconventional reservoirs. The drilling environment is becoming more and more complex. However， in the process of drilling， under the effects of the drilling fluid in the borehole， the pore pressure and temperature can change， which causes stress redistribution of the borehole， and the stability of the borehole can be affected， it is easily causing well blowing， mud loss， borehole collapse and sticking accidents. Scholars at home and abroad have carried out a more comprehensive study on borehole stability. The borehole stability mechanism has been studied from fluid-solid coupling， thermal-flow-solid coupling and mechanics-chemical coupling. Xinglong Wang and Yuanfang Cheng[7] have established borehole wall temperature and pressure coupling mechanical model in shale formations， illustrating that temperature changes can cause the variation of pore pressure， changing the borehole wall stress distribution by numerical method. Baohua Wei[8] has analyzed the effect of temperature and pressure on borehole stability， establishing additional stress calculation model caused by pore pressure changes. Lewen Zhang[9] has established mechanics-chemical coupling mathematical model for calculating borehole wall stress， the wellbore wall pore pressure and stress variation regulations have been analyzed under the effect of mechanics-chemical coupling. Although some scholars have discussed the effect of pore pressure and temperature difference variation on borehole stability， the theory of effects of pore pressure and temperature changes on porosity and wall stress is limited， and the borehole stability model considering simultaneous changes of pore pressure and temperature has not been reported. In this paper， the problem has been better explained in theory， which is a supplement and completeness for the existing borehole stability mechanism. 1. THE MECHANICAL MODEL OF BOREHOLE ROCK
In the process of drilling， the mechanical model of borehole rock is shown in Figure 1. Assuming that formation rock is isotropic elastic medium， the horizontal maximum principal stressand minimum principal stressare loaded on formation rock. pi is the pressure caused by drilling fluid column in borehole， pp is the initial formation pore pressure.
In the process of drilling， the key factor affects borehole stability is the variation of formation temperature and pressure. On the one hand， in order to prevent well blowing， mud loss， it is required that the drilling fluid column pressure is greater than the pore pressure， which can lead to drilling fluid filtrate seep into formation， and it can generate percolation zone around the borehole （gray area within the dotted line in Figure 1）， and the solid phase particles in drilling fluid are left on the borehole wall in the forms of mud cake （the black ring region in Figure 1）. On the other hand， the temperature difference between the deep formation and fluid can generate additional stress and strain on borehole wall rock on the basis of the original balance， which can change the original bearing force balance.
2. THE BOREHOLE WALL STRESS DISTRIBUTION CONSIDERING PORE PRESSURE AND TEMPERATURE DIFFERENCE VARIATION
Pore pressure and temperature difference variation will cause the wall stress distribution variation. Considering in situ stress， drilling fluid column pressure， fluid seepage and thermal stress effects separately that can cause borehole wall stress variation， borehole wall stress is analyzed.
2.1 Borehole Wall Stress Distribution Caused by in Situ Stress
When the borehole is opened， the stress concentration can generate around the borehole wall under in situ stress field， wall rock stress state can be expressed as follow[10].
Where： σr is radial stress of the wall rock， MPa； σθ is circumferential stress of the wall rock， MPa； σz is the vertical stress of the wall rock， MPa； σH is the horizontal maximum principal stress， MPa； σh is the horizontal minimum principal stress， MPa； σv is overburden stress， MPa； r is the distance between any point on the borehole wall rock and the center of the borehole， m； R is the radius of the borehole， m； θ is the included angle between any point on the borehole wall rock and the horizontal maximum principal stress， （°）； is the poisson ratio of the rock.
2.2 Borehole Wall Stress Caused by Drilling Fluid Column Pressure and Seepage Effects Under the effects of drilling fluid column pressure and the additional stress caused by fluid seepage， the stress distribution caused by wall surrounding area can be expressed as follow[10].
where： Pi is drilling fluid column pressure in borehole， MPa； Pp is formation initial pore pressure， MPa； α is effective stress factor； is the porosity of the wall rock.
2.3 The Thermal Stress Analysis of the Borehole Wall
The borehole wall stress variation caused by the temperature difference between drilling fluid and borehole rock can be expressed as follow.
Where： E is Young’s modulus of borehole rock， MPa； T is the temperature of borehole rock that has change， （℃）； T0 is the initial temperature of borehole rock， （℃）.
2.4 The Porosity Variation of Borehole Rock
In the process of drilling， the porosity of rock will change under the effect of pore pressure and thermal stress. By changing the porosity that has changed can be expressed as follow.
Where： is the porosity of the rock that has changed； is the initial porosity of the rock； is the pore pressure variation difference caused by drilling fluid filtration， MPa.
2.5 The Stress Distribution of Borehole Wall
The rock stress distribution can be obtained by the superposition of the various parts of the stress according to linear superposition theory in elastic mechanics theory.
The effects of pore pressure and temperature difference of the rock on borehole rock stress can be obtained by analyzing the above equation， then study their effects on safety mud density window by analyzing stress variation of the borehole wall.
3. THE DETERMINATION OF SAFE MUD DENSITY WINDOW
The different stress states can be obtained by borehole wall stress distribution model. The caving pressure can be obtained by deriving Moore-Coulomb criterion. The breakdown pressure of the borehole wall can be gotten according to the tensile failure criterion. The safety mud density window calculating model is established considering fluid seepage and temperature variation on the condition that the overburden stress is the intermediate principal stress.
By deriving the drilling fluid density that is equivalent to the caving pressure can be expressed as follow.
Where： ρm is the drilling fluid density that is equivalent to the caving pressure， g/cm3；， is the angle of internal friction of the rock， （°）； is the well depth， m； C0 is cohesive force of the rock， MPa. The drilling fluid density that is equivalent to the breakdown pressure can be expressed as follow.
Where： Pf is the drilling fluid density that is equivalent to the breakdown pressure， g/cm3； is the tensile strength of the rock， MPa.
4. THE EFFECTS OF PORE PRESSURE AND TEMPERATURE DIFFERENCE VARIATION ON BOREHOLE STABILITY
In order to analyze the effects of pore pressure and temperature difference variation on borehole stability， the rock mechanics parameters are obtained from the rock at the depth of 2400 m. The horizontal maximum principal stress is 48 MPa. The horizontal minimum principal stress is 38 MPa. Poisson’s ratio is 0.26. The Young’s modulus is 13700 MPa. The porosity is 0.2. The tensile strength of rock is 3.55 MPa. The cohesive force is 6.33 MPa. The angle of internal friction is 33.11°. The effective stress coefficient is 0.7. The thermal expansion coefficient is 0.00005. The initial formation pore pressure is 23.86 MPa.
The effects of borehole rock pore pressure and temperature difference variation on safe mud density window can be shown in Figure 2 and Figure 3. The result can be shown as follow： （1） When the temperature difference is constant， with the enhancement of fluid filtration， borehole rock pore pressure increasing， causing the collapse pressure increasing， breakdown pressure decreasing， the safe drilling fluid density window becoming smaller， and it is not conducive to drilling safely. Thus， for a narrow safe density window drilling， the building capacity of the fluid should be strengthened， reducing effects of fluid filtration on wellbore stability. （2） When the borehole wall rock pore pressure is constant， if the wall rock temperature increases， with the temperature difference increasing， both the collapse pressure and breakdown pressure increasing， safe drilling fluid density window becoming larger， it is conducive to drilling safely. If wall rock temperature is lowered， with the temperature difference increasing， both the collapse pressure and breakdown pressure decreasing， the safety drilling fluid density window becoming smaller， it is not conducive to drilling safely. It can be shown that the cooling effect of drilling fluid to the formation is not conducive to borehole stability.
CONCLUSION
When the temperature difference between borehole rock and drilling fluid is constant， with the enhancement of fluid filtration， borehole rock pore pressure increasing， causing the collapse pressure increasing， breakdown pressure decreasing， the safe drilling fluid density window becoming smaller， borehole stability becoming deterioration， it is not conducive to drilling safely. When the borehole wall rock pore pressure is constant， if drilling fluid makes wall rock temperature decreasing， with the temperature difference increasing， both the collapse pressure and breakdown pressure decreasing， the safety drilling fluid density window becoming smaller， borehole stability becoming deterioration， it is not conducive to drilling safely. If drilling fluid make wall rock temperature increasing， with the temperature difference increasing， both the collapse pressure and breakdown pressure increasing， safe drilling fluid density window becoming larger， the borehole wall tending to stabilize， it is conducive to drilling safely. REFERENCES
[1] Aadnow， B. S.， Rogaland， U.， & Chenevert， M.E. （1987）. Stability of Highly Inclined Boreholes. SPE 16052.
[2] Yew， C. H. （1989）. On Fracturing Design of a Deviated Wells. SPE 19722.
[3] Jia， G. Y.， Zhang， L. S.， & Duan， Y. X. （2006）. Drilling in High Density and Thin Pressure Windows Complex Formation. SPE 104411.
[4] Wang， Q.， Zhou， Y. C.， Wang， G.， et al. （2012）. The Flow-Solid-Thermal Coupling Model for Shale Borehole Stability. Petroleum Exploration and Development， 39（4）， 475-480.
[5] Li， Y. F.， Fu， Y. Q.， Tang， G.， et al. （2012）. The Regulations of Stress Type Affecting Borehole Stability for Directional Wells. Natural Gas Industry， 32（3）， 78-80.
[6] Li， H. F.， & Chen， M. （2011）. Real-Time Prediction of Borehole Instability Based on Actual Drilling Data. Acta Petrolei Sinica， 32（2）， 324-328.
[7] Wang， X. L.， Cheng， Y. F.， & Zhao， Y. Z. （2007）. The Effect of Temperature on Wellbore Stability in Shale During Drilling. Petroleum Drilling Techniques， 35（2）， 42-45.
[8] Wei， B. H.， Lu， X. F.， Wang， B. Y.， et al. （2004）. The Effect of Formation Temperature Variation on Wellbore Stability in High Temperature Wells. Drilling Fluid and Completion Fluid， 21（6）， 15-18.
[9] Zhang， L. W.， Qiu， D. H.， & Cheng， Y. F. （2009）. Research on the Wellbore Stability Model Coupled Mechanics and Chemistry. Journal of Shandong University （Engineering Science）， 39（3）， 111-114.
[10] Chen， M.， Jin， Y.， & Zhang， G. Q. （2008）. Petroleum Engineering Rock Mechanics （pp. 62-65）. Beijing： Science Press.