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A Coupled Mechanical-Thermal-Physico-Chemical Model for the Study of Time-Dependent Wellbore Stability in Shales
581 A COUPLED MECHANICAL-THERMAL-PHYSICO-CHEMICAL MODEL FOR THE STUDY OF TIME-DEPENDENT WELLBORE STABILITY IN SHALES Choi. S.K., Tan, C.P. and Freij-Ayoub, R CSIRO Petroleum, Australia Abstract: The analysis of wellbore stability in shales needs to take into account the interaction of the drilling fluid and formation rock. Depending on the borehole trajectory, formation rock properties, in-situ stress, mud pressure, type of drilling fluid and dissolved ions, and temperature of drilling fluid and formation, the interaction may involve invasion of mud filtrate into the formation, single- or two-phase flow of formation pore fluid, transport and diffusion of salt ions (solutes), mechanical deformation caused by anisotropic swelling of shales, thermal stress, poroelastic effects, and possibly failure of the formation rock. Such failure can be almost instantaneous or time-dependent. To solve the coupled partial differential equations describing the complex interaction of the various processes and mechanisms, a finite element program has been developed to specifically model the mechanical-thermal-physico-chemical interactions associated with wellbore stability in shales. ]n the model, the shale is treated as a transversely isotropic medium. A brief description of the models and the underlying theories are presented in this paper. Some model results are presented and discussed.
1, INTRODUCTION Wellbore stability in shales is strongly influenced by the evolution of the effective stress field around the wellbore as a result of the interaction of a number of factors and processes (See for example Tan et al., 1998; Choi and Tan, 1998). The important factors include orientation of the wellbore, magnitude and orientation of the in-situ principal stresses, properties of the bedding planes and other major geological structures, type of drilling fluid and dissolved ions, mud pressure, formation pore pressure, temperature of drilling fluid and formation rock, and the fluid transport, thermal and geomechanical properties of the shale, and thermoporoelastic effects. These factors can have an important influence on processes such as pore fluid flow and solute transport induced by the difference in pressure and solute concentration between the drilling fluid and formation, invasion of formation by mud filtrate, swelling of shale due to change in water potential, thermal diffusion caused by difference in temperature between drilling fluid and formation, and induced thermal expansion of pore fluid and shale. These processes and their interaction can have significant influence on the evolution of the effective stress field around the wellbore. Some of the processes are strongly coupled and must be treated accordingly. A coupled finite element program has been developed to model these processes and the effects of their interaction on time-dependent wellbore stability.
2. MAJOR PROCESSES AND MECHANISMS The processes can be considered as three initially independent processes. These include the undrained load-deformation-failure process and the fluid and heat flow processes. The fluid flow process may include solute transport. Other mechanisms include swelling of shale caused by change in water potential resulting from the other processes. The main coupling parameters are stress, pore pressure and temperature.
2.1 Load'deformation-fallure process The borehole is assumed to have a long straight section, which can be vertical or inclined, and the borehole axis may not align with any of the principal in-situ stresses or with the direction normal to the bedding planes. Under some conditions, crosssectional planes which are initially perpendicular to the borehole axis before drilling may no longer be planar after creation of the borehole and a 3dimensional model is required. To reduce the computational effort, especially for coupled analyses, a generalised plane strain (Leknitskii, 1981; Aoki, 1994) instead of a fully 3-dimensional modelling approach is adopted. Due to the relatively high porosity and low permeability, fluid saturated pores and the existence of bedding planes in most shales, it is necessary to take into account the effects of material anisotropy and poroelastic effects (Biot, 1956) on wellbore stability in such formations. The constitutive equation for a transversely isotropic poroelastic material can be found, for example, in Cheng (1997).
582 2.2 Fluid flow process Loss of confinement resulting from the creation of a borehole can lead to wellbore failure. The weight of the drilling fluid provides some of the support (for the wellbore) which was originally provided by the drilled out material. However, when drilling under an overbalance condition in shales without an effective flow barrier present at the wellbore wall, invasion of the mud filtrate into the formation may occur. Due to the saturation and low permeability of shales, a small volume of mud filtrate penetrating the formation will result in a considerable increase in pore pressure near the wellbore wall. The increase in pore pressure reduces the effective mud support and can lead to a less stable wellbore condition. The penetration of mud filtrate will depend on the wettability of the drilling fluid and pore fluid to the shale, miscibility of the drilling fluid and pore fluid, and geometrical properties of the inter-connected pores within the formation. If the drilling fluid is a non-wetting fluid, invasion of the mud filtrate into the formation will only occur if the mud pressure exceeds the formation pressure by at least a threshold value referred to as breakthrough pressure (Tan et al., 1996a) given by:
where ^ is porosity, S is pore diameter parameter and T is parameter similar to tortuosity. The parameter S in the above equation can be determined from the differential pore size distribution, a(6):
J a(S) =
(2)
df(6) d5
(4)
where f(8) is cumulative pore size distribution. Using Equation 2, the parameter T for a rock material can be determined from the measured absolute permeability and porosity, and the parameter calculated from the cumulative pore size distribution of the rock material. The modified form of Equation 2 for the apparent permeability, kapp is given by:
k_ =
k = '-
(3)
The differential pore size distribution is defined as
p,,=p.-p„-p,=j[{s.cosel^-{s,cose)J (1) where Pbt is breakthrough pressure, Pc is capillary pressure, ?„, is mud pressure, Pf is formation pressure, St is surface tension, 9 is contact angle (fluid-air contact), 6 is pore diameter and subscripts mf and pf refer to mud filtrate and pore fluid respectively. If the fluids are immiscible, a pressure difference equal to the capillary pressure of the two fluids exists permanently across the mud filtrate-pore fluid interface when flow occurs. Depending on the size of the pores, the mud filtrate may not be able to penetrate the smaller pores. The apparent permeability, a function of the proportion of pores penetrated by the mud filtrate, can be determined from the pore size distribution curve, absolute permeability and porosity of the rock material (Tan et al., 1996a). The methodology is based on a simple capillaric model which represents a porous medium by a bundle of parallel capillaries (Scheideggar, 1974). From the law of HagenPoiseuille, the absolute permeability of such a model is given by:
S'a{S)dS
f(5)(t)5,
(5)
where 5 app is an apparent pore diameter parameter. For a water-based drilling fluid, the front of the filtrate which penetrated into the pores of the rock would gradually mix with the pore fluid. As a consequence, the threshold breakthrough pressure will decrease with time as a result of the reduction in interfacial tension between the two fluids.
2.3 Solute and fluid flow induced by osmosis - chemical potential mechanism The fine pores and negative charge of clay on pore surfaces make shales exhibit membrane behaviour (Whitworth and Fritz, 1994). Hence, the flow of water out of (or into) shales due to the chemical potential mechanism is somewhat similar to the flow of water through a semi-permeable membrane (Mody and Hale, 1993; van Oort et al., 1995; Tan et al., 1996b). The driving force involved in the water transportation (for zero overbalance condition) is the chemical potential gradient across the membrane that is generally related to the difference in solute (salt) concentration (water activity) (Tan et al., 1996b):
583 ^
= CjV(KVO)]
(6)
where 0 is rcx:k aqueous potential, t is time, CQ is —— = 1/(M^-f (t)/K^), 9w is volumetric water content. My is coefficient of volume change of rock (l/(Kr + 4/3Gr)), Kf is bulk modulus of rock, G^ is shear modulus of rock, (|) is rock porosity, K^ is bulk modulus of water, K is hydraulic conductivity of rock = k/)Li, k is absolute permeability, i^ is fluid viscosity which is a function of salt concentration, salt type and temperature, and V is mathematical operator gradient (5/5x,8/6y,5/6z). The solute flow process includes advection (or convection), dispersion and diffusion. The equation for the process can be written as
at
-v(c • q - e^D, • vc - e^D • vc)(i - a,)
(7)
where ^ is rock porosity, q is water flux, t is time, C is solute concentration of pore fluid, Dg is coefficient of dispersion, D is coefficient of molecular diffusion, a^ is reflection coefficient which describes the solute leakage through the membrane system, and 9w is volumetric water content. The osmotic pressure for a non-ideal semipermeable membrane is given by
(8) RT M^ ^^^ ^
it has a range of pore sizes including wide pore throats that result in significant permeability to solutes. The wide throats reduce the solute interaction with the pore surfaces which increases the permeability of the membrane to solutes. The reflection coefficient (a,-) and osmotic pressure coefficient (Op) would only be the same for either a fully permeable membrane or an ideal semi-permeable membrane. The reflection coefficient (a^) can be determined from either empirical relationships with osmotic pressure coefficient based on back-analyses of the experimental results obtained for a given drilling fluid-shale system or theoretical equations based on thermodynamics (van Oort et al., 1995; ZeynalyAndabily et al., 1996).
2.4 Thermoporoelasticity mechanism The drilling fluid often has a temperature different from the formation as a result of thermal gradient down the borehole. The temperature difference between the drilling fluid and the formation will result in heat transfer between the two media leading to change in pore pressure due to thermoporoelastic effects (Choi and Tan, 1998). Due to the low permeability of shales, the coefficient of thermal diffusivity is at least a few orders of magnitude greater than the coefficient of fluid diffusivity. Hence, heat transfer in the formation will be dominated by diffusion, and convective transfer by fluid flow may be ignored. Since the coefficient of thermal expansion of pore fluid is much larger (in the order of 1(X) times) than the coefficient of rock solid, temperature change will result in a change in pore pressure. The governing equations for conductive heat transfer are given by: V(CoVT) = T
(9)
p V M. where (p is osmotic pressure, oCp is osmotic pressure coefficient, R is gas constant, T is absolute temperature (°K), V is partial molar volume of water, A^ is water activity, M^ is molecular weight of solvent, Mg is molecular weight of salt, V^ is valency of salt and CQ is osmotic coefficient of salt in the solution. For an ideal semi-permeable membrane, all solutes are reflected by the membrane and only water molecules can pass through the membrane. However, shale exhibits a non-ideal semi-permeable ('leaky') membrane behaviour to water-based solutions because
""^'ps where c^ is thermal diffusivity, X is thermal conductivity, p is mass density, s is specific heat (or •
r)T
thermal capacity), T is temperature and T = —- . at The effects of temperature change on pore fluid pressure (Wang et al., 1996) is given by:
cV^p = p-c'T
(10)
584
2.5 Swelling and hydrational stress mechanism
2KGB^(I + V„)P ( 1 - V )
20^5^-^^)
fi(l-v)(l + v j
^(p{af-a,)
where c is generalised consolidation coefficient, K is k - , k is intrinsic permeability, \x is viscosity of pore fluid, G is shear modulus, v is drained Poisson's ratio, Vy is undrained Poisson's ratio, B is Skempton's pore pressure coefficient, a^ is coefficient of thermal expansion of solid matrix, af is coefficient of thermal expansion of pore fluid and (J> is porosity. The pore fluid flow is described by: d0 = c[v(/r.V0)] dt
(11)
where
3(1-2v)
a
ATS,
The total aqueous potential (pressure and chemical potential) of the pore fluid increases with the increase in pore pressure and/or chemical potential (decrease in salt concentration). When the total aqueous potential of the pore fluid increases, water will be adsorbed into the clay platelets. The water adsorption will result in either the platelets moving further apart i.e., swelling if they are free to expand, or the generation of hydrational stress if swelling is constrained. The process can be represented by (Tan et al., 1997): Ae^ =QiAlogO
(13)
where £« is strain due to change in rock water potential in direction i, Qi is volume change parameter at a given stress in direction i, and O is rock water potential. It is assumed that rock water potential will affect the rock matrix (effective) stress, and therefore the change in swelling strain can also be expressed as: Ae . =C.Aa.
(14)
where Q is volume change coefficient at a given stress in direction i, and a/ is effective stress in direction i. The volume change coefficient can be obtained from the following equation C; =
(15)
The generation of hydrational stress due to the rock material constrained from swelling is given by (12)
K a =l- — Ks where 5ij is Kronecker delta, K is drained bulk modulus of rock framework, and Kg is bulk modulus of mineral constituents. If a transient temperature field exists in the saturated rock, the difference in coefficient of thermal expansion of the pore fluid and the rock matrix can induce changes in pore pressure and stresses. This can lead to thermal-induced pore fluid flow and deformation of the rock around the borehole.
Ao^i=[D]^a^
(16)
where Goi is hydrational stress in direction i; and ID] is stiffness matrix.
3. INTEGRATION OF THE PROCESSES AND MECHANISMS The model takes into account changes in formation pore pressure arising from fluid flow and induced by deformation of the formation rock, and changes in effective stresses as a result of change in the formation pore pressure. In the mud pressure penetration model, the geometry of the pore network was represented as a series of parallel cylindrical tubes. Although the model
585 may not represent the actual flow processes in the pore network at microscopic scale, the approach enables certain properties of the porous medium (or equivalent continuum) for certain flow process at the macroscopic scale to be derived. The main effect of the mud pressure penetration process is in modifying the effective permeability. Chemical potential difference will affect the flow rate of dissolved solute ions and solvent (or pore fluid - the mud filtrate is considered to be pore fluid once it has infiltrated the pores of the shale). The main consideration of dissolved ions is their contribution to osmotic pressure. This is achieved by keeping track of the concentration of the dissolved ions (Equation 7). The effect of osmotic pressure on the flow of pore fluid is taken into account by including the osmotic pressure in the rock water potential, and use the spatial gradient of rock water potential (Equation 6) as the driving force for pore fluid flow. The effects of temperature can be prevalent for some formation rocks and in-situ conditions. Temperature-induced changes in pore pressure and rock matrix stress (thermo-poro-elasticity model) can be modelled using Equations 10 to 12. hi addition, to take into account the chemical potential effects, gradient of rock water potential instead of pore pressure gradient is used in Equation 11. As the effects of (rock matrix) swelling are mainly on effective stress, they can be taken into account by keeping track of changes in rock water potential or effective stresses, and the strain induced (swelling and hydrational stress model. Equations 14 to 16). The hydrational stress can be considered as additional internal stress in the equilibrium equations. For all the processes and mechanisms, the major coupling parameters are stress, pore fluid pressure and temperature.
dependent because some of the processes such as mud filtrate invasion, fluid flow and thermal diffusion are time-dependent. The results also show that thermoporoelastic ity effects can induce a pore pressure change of about 0.7 MPa close to the wellbore wall for every 1°C temperature difference between the drilling fluid and formation. The model results also show that the chemical potential mechanism can improve wellbore stability by lowering the pore pressure below the initial value in the formation near to the wellbore.
I 30 t-0 days t-0.01 days t-0.4 days t-2.5 days
1
1.5 2 Normalised distance from weHbore centre
2.5
Figure 1. The effects of coupled interaction of mechanical deformation, fluid flow & a colder drilling fluid on pore pressure.
4. MODEL RESULTS AND DISCUSSIONS Analyses have been conducted to study the effects of the different processes and their interaction on wellbore stability. These include coupling of some or all of the processes and mechanisms. Examples of the model results are shown in Figures 1 and 2. Figure 1 shows the change in pore pressure at different distance (normalized with respect to wellbore radius) from the wellbore center with time when drilling in a formation which is 30°C hotter than the drilling fluid, taking into account the effects of borehole deformation, mud filtrate invasion, thermal diffusion and thermoporoelastic effects. Figure 2 shows the model results which take into account borehole deformation, mud filtrate invasion, fluid flow, solute transport, chemical potential mechanism, and poroelastic effects. The model results show that stability can be time-
^lOdays ^•0.01 days • M).4days t-2.5 days
1
1.5 2 2.5 3 Nomfialised distancefromwellbore centre
Figure 2. The effects of coupled interaction of mechanical deformation, fluid flow & a lower water activity drilling fluid on pore pressure.
586 5. CONCLUSION A finite element program has been developed which can model the effects of the interaction of a number of important factors and processes on wellbore stability in shales. The factors and processes which have been incorporated into the model include the orientation of the wellbore, magnitude and orientation of the in-situ principal stresses, properties of the bedding planes, properties of drilling fluid and dissolved ions, mud pressure, formation pore pressure, temperature of drilling fluid and formation rock, and the fluid transport, thermal, geomechanical and properties of shale, swelling and hydrational stress mechanism, and thermoporoelastic effects. The results of some preliminary modelling studies showed that, as some of the important field variables such as formation pore pressure, rock aqueous potential, stress and temperature will change with time before a steady state condition is reached, wellbore stability in shales can be time-dependent and this must be taken into consideration in drilling fluid design and shale instability management. The models results in this paper show that the temperature or chemical potential of the drilling fluid can have significant influence on how the pore pressure field in the formation adjacent to the wellbore wall may change with time as a result of heat diffusion, thermoporoelastic effect, and chemical potential mechanism. Results of the modelling study conducted to date indicate that, in order to predict both short and long term stabilities of wellbore in shales, it is important to take into account the interaction of the processes and mechanisms described in this paper. The coupled model provides a tool for studying the relative importance of the various factors and processes on wellbore stability in shales, and how stability may change with time.
REFERENCES Aoki, T. (1994). Stress-strain-pore pressure interaction for anisotropic porous rock, Taisei Technical Research Report, 27, 359-366. Biot, M.A. (1956). General solutions of the equations of elasticity and consolidation for a porous material. J. Appl. Mech., Trans. Am. Soc. Mech. Engrs.,Vol. 78,pp. 91-96. Cheng, A.H.-D. (1997). Material coefficients of anisotropic poroelasticity. Int. J. Rock Mech. Min. Sci., Vol. 34, No. 2, pp. 199-205. Choi, S.K. and Tan, C.P. (1998). Modelling of effects of drilling fluid temperature on wellbore stability. SPE 47304, Proc. SPE/ISRM EUROCK' 98, Trondheim, Norway, 8-10 July 1998, Vol. 1, pp. 471-477.
Leknitskii, S.G. (1981). Theory of Elasticity of an Anisotropic Body. Mir Publ. Mody, F.K. and Hale, A.H. (1993). A borehole stability model to couple the mechanics and chemistry of drilling fluid interaction, Proc. SPE/IADC Drilling Conf., Amsterdam, The Netherlands, 473-490 Scheidegger, A.E. (1974). The Physics of Flow Through Porous Media, 1974, Univ of Toronto Press, Britain. Tan, C.P., S.S. Rahman, B.C. Richards and F.K. Mody (1998) Integrated Approach to Drilling Fluid Optimization for Efficient Shale Instability Management. Proc. SPE Int. Oil and Gas Conf. and Exhibition in China, Beijing, China. Tan, C.P., Richards, C.G., Rahman, S.S. and Andika, R. (1997). Effects of swelling and hydrational stress in shales on wellbore stability. Asia Pacific Oil and Gas Conference and Exhibition, Kuala Lumpur, Malaysia, 345-349. Tan, C.P., E.M. Zeynaly-Andabily and S.S. Rahman (1996a). A Novel Method of Screening Drilling Muds Against Mud Pressure Penetration for Effective Borehole Wall Support. lADC/SPE Asia Pacific Drilling Technology Conf., Kuala Lumpur, Malaysia. Tan, C.P., B.G Richards and S.S. Rahman.( 1996b). Managing Physico-Chemical Wellbore Instability in Shales with the Chemical Potential Mechanism. SPE Asia Pacific Oil and Gas Conf., Adelaide, Australia. (1996b). van Oort, E., Hale, A.H. and Mody, F.K. (1995). Manipulation of coupled osmotic flows for stabilisation of shales exposed to water-based drilling fluids. Proc. 70th SPE Annual Technical Conference and Exhibition, Dallas, USA, 497-509. Wang, Y., Papamichos, E. and Dusseault, M.B. (1996). Thermal Stresses and Borehole Stability in Rock Mechanics. Proc. 2nd NARMS, Rock Mechanics Tools and Techniques, pp. 1121-1126. Whitworth, T.M. and Fritz, S.J. (1994). Electrolyteinduced solute permeability effects in compacted smectite membranes. Applied Geochemistry, 533546. Zeynaly-Andabily, E.M., Chen, H., Rahman, S.S. and Tan, C.P. (1996). SPE 36396 Management of wellbore instability in shales by controlling the physical-chemical properties of muds. SPE Asia Pacific Drilling Technology Conf., Kuala Lumpur, Malaysia, pp. 253-261.
1, INTRODUCTION Wellbore stability in shales is strongly influenced by the evolution of the effective stress field around the wellbore as a result of the interaction of a number of factors and processes (See for example Tan et al., 1998; Choi and Tan, 1998). The important factors include orientation of the wellbore, magnitude and orientation of the in-situ principal stresses, properties of the bedding planes and other major geological structures, type of drilling fluid and dissolved ions, mud pressure, formation pore pressure, temperature of drilling fluid and formation rock, and the fluid transport, thermal and geomechanical properties of the shale, and thermoporoelastic effects. These factors can have an important influence on processes such as pore fluid flow and solute transport induced by the difference in pressure and solute concentration between the drilling fluid and formation, invasion of formation by mud filtrate, swelling of shale due to change in water potential, thermal diffusion caused by difference in temperature between drilling fluid and formation, and induced thermal expansion of pore fluid and shale. These processes and their interaction can have significant influence on the evolution of the effective stress field around the wellbore. Some of the processes are strongly coupled and must be treated accordingly. A coupled finite element program has been developed to model these processes and the effects of their interaction on time-dependent wellbore stability.
2. MAJOR PROCESSES AND MECHANISMS The processes can be considered as three initially independent processes. These include the undrained load-deformation-failure process and the fluid and heat flow processes. The fluid flow process may include solute transport. Other mechanisms include swelling of shale caused by change in water potential resulting from the other processes. The main coupling parameters are stress, pore pressure and temperature.
2.1 Load'deformation-fallure process The borehole is assumed to have a long straight section, which can be vertical or inclined, and the borehole axis may not align with any of the principal in-situ stresses or with the direction normal to the bedding planes. Under some conditions, crosssectional planes which are initially perpendicular to the borehole axis before drilling may no longer be planar after creation of the borehole and a 3dimensional model is required. To reduce the computational effort, especially for coupled analyses, a generalised plane strain (Leknitskii, 1981; Aoki, 1994) instead of a fully 3-dimensional modelling approach is adopted. Due to the relatively high porosity and low permeability, fluid saturated pores and the existence of bedding planes in most shales, it is necessary to take into account the effects of material anisotropy and poroelastic effects (Biot, 1956) on wellbore stability in such formations. The constitutive equation for a transversely isotropic poroelastic material can be found, for example, in Cheng (1997).
582 2.2 Fluid flow process Loss of confinement resulting from the creation of a borehole can lead to wellbore failure. The weight of the drilling fluid provides some of the support (for the wellbore) which was originally provided by the drilled out material. However, when drilling under an overbalance condition in shales without an effective flow barrier present at the wellbore wall, invasion of the mud filtrate into the formation may occur. Due to the saturation and low permeability of shales, a small volume of mud filtrate penetrating the formation will result in a considerable increase in pore pressure near the wellbore wall. The increase in pore pressure reduces the effective mud support and can lead to a less stable wellbore condition. The penetration of mud filtrate will depend on the wettability of the drilling fluid and pore fluid to the shale, miscibility of the drilling fluid and pore fluid, and geometrical properties of the inter-connected pores within the formation. If the drilling fluid is a non-wetting fluid, invasion of the mud filtrate into the formation will only occur if the mud pressure exceeds the formation pressure by at least a threshold value referred to as breakthrough pressure (Tan et al., 1996a) given by:
where ^ is porosity, S is pore diameter parameter and T is parameter similar to tortuosity. The parameter S in the above equation can be determined from the differential pore size distribution, a(6):
J a(S) =
(2)
df(6) d5
(4)
where f(8) is cumulative pore size distribution. Using Equation 2, the parameter T for a rock material can be determined from the measured absolute permeability and porosity, and the parameter calculated from the cumulative pore size distribution of the rock material. The modified form of Equation 2 for the apparent permeability, kapp is given by:
k_ =
k = '-
(3)
The differential pore size distribution is defined as
p,,=p.-p„-p,=j[{s.cosel^-{s,cose)J (1) where Pbt is breakthrough pressure, Pc is capillary pressure, ?„, is mud pressure, Pf is formation pressure, St is surface tension, 9 is contact angle (fluid-air contact), 6 is pore diameter and subscripts mf and pf refer to mud filtrate and pore fluid respectively. If the fluids are immiscible, a pressure difference equal to the capillary pressure of the two fluids exists permanently across the mud filtrate-pore fluid interface when flow occurs. Depending on the size of the pores, the mud filtrate may not be able to penetrate the smaller pores. The apparent permeability, a function of the proportion of pores penetrated by the mud filtrate, can be determined from the pore size distribution curve, absolute permeability and porosity of the rock material (Tan et al., 1996a). The methodology is based on a simple capillaric model which represents a porous medium by a bundle of parallel capillaries (Scheideggar, 1974). From the law of HagenPoiseuille, the absolute permeability of such a model is given by:
S'a{S)dS
f(5)(t)5,
(5)
where 5 app is an apparent pore diameter parameter. For a water-based drilling fluid, the front of the filtrate which penetrated into the pores of the rock would gradually mix with the pore fluid. As a consequence, the threshold breakthrough pressure will decrease with time as a result of the reduction in interfacial tension between the two fluids.
2.3 Solute and fluid flow induced by osmosis - chemical potential mechanism The fine pores and negative charge of clay on pore surfaces make shales exhibit membrane behaviour (Whitworth and Fritz, 1994). Hence, the flow of water out of (or into) shales due to the chemical potential mechanism is somewhat similar to the flow of water through a semi-permeable membrane (Mody and Hale, 1993; van Oort et al., 1995; Tan et al., 1996b). The driving force involved in the water transportation (for zero overbalance condition) is the chemical potential gradient across the membrane that is generally related to the difference in solute (salt) concentration (water activity) (Tan et al., 1996b):
583 ^
= CjV(KVO)]
(6)
where 0 is rcx:k aqueous potential, t is time, CQ is —— = 1/(M^-f (t)/K^), 9w is volumetric water content. My is coefficient of volume change of rock (l/(Kr + 4/3Gr)), Kf is bulk modulus of rock, G^ is shear modulus of rock, (|) is rock porosity, K^ is bulk modulus of water, K is hydraulic conductivity of rock = k/)Li, k is absolute permeability, i^ is fluid viscosity which is a function of salt concentration, salt type and temperature, and V is mathematical operator gradient (5/5x,8/6y,5/6z). The solute flow process includes advection (or convection), dispersion and diffusion. The equation for the process can be written as
at
-v(c • q - e^D, • vc - e^D • vc)(i - a,)
(7)
where ^ is rock porosity, q is water flux, t is time, C is solute concentration of pore fluid, Dg is coefficient of dispersion, D is coefficient of molecular diffusion, a^ is reflection coefficient which describes the solute leakage through the membrane system, and 9w is volumetric water content. The osmotic pressure for a non-ideal semipermeable membrane is given by
(8) RT M^ ^^^ ^
it has a range of pore sizes including wide pore throats that result in significant permeability to solutes. The wide throats reduce the solute interaction with the pore surfaces which increases the permeability of the membrane to solutes. The reflection coefficient (a,-) and osmotic pressure coefficient (Op) would only be the same for either a fully permeable membrane or an ideal semi-permeable membrane. The reflection coefficient (a^) can be determined from either empirical relationships with osmotic pressure coefficient based on back-analyses of the experimental results obtained for a given drilling fluid-shale system or theoretical equations based on thermodynamics (van Oort et al., 1995; ZeynalyAndabily et al., 1996).
2.4 Thermoporoelasticity mechanism The drilling fluid often has a temperature different from the formation as a result of thermal gradient down the borehole. The temperature difference between the drilling fluid and the formation will result in heat transfer between the two media leading to change in pore pressure due to thermoporoelastic effects (Choi and Tan, 1998). Due to the low permeability of shales, the coefficient of thermal diffusivity is at least a few orders of magnitude greater than the coefficient of fluid diffusivity. Hence, heat transfer in the formation will be dominated by diffusion, and convective transfer by fluid flow may be ignored. Since the coefficient of thermal expansion of pore fluid is much larger (in the order of 1(X) times) than the coefficient of rock solid, temperature change will result in a change in pore pressure. The governing equations for conductive heat transfer are given by: V(CoVT) = T
(9)
p V M. where (p is osmotic pressure, oCp is osmotic pressure coefficient, R is gas constant, T is absolute temperature (°K), V is partial molar volume of water, A^ is water activity, M^ is molecular weight of solvent, Mg is molecular weight of salt, V^ is valency of salt and CQ is osmotic coefficient of salt in the solution. For an ideal semi-permeable membrane, all solutes are reflected by the membrane and only water molecules can pass through the membrane. However, shale exhibits a non-ideal semi-permeable ('leaky') membrane behaviour to water-based solutions because
""^'ps where c^ is thermal diffusivity, X is thermal conductivity, p is mass density, s is specific heat (or •
r)T
thermal capacity), T is temperature and T = —- . at The effects of temperature change on pore fluid pressure (Wang et al., 1996) is given by:
cV^p = p-c'T
(10)
584
2.5 Swelling and hydrational stress mechanism
2KGB^(I + V„)P ( 1 - V )
20^5^-^^)
fi(l-v)(l + v j
^(p{af-a,)
where c is generalised consolidation coefficient, K is k - , k is intrinsic permeability, \x is viscosity of pore fluid, G is shear modulus, v is drained Poisson's ratio, Vy is undrained Poisson's ratio, B is Skempton's pore pressure coefficient, a^ is coefficient of thermal expansion of solid matrix, af is coefficient of thermal expansion of pore fluid and (J> is porosity. The pore fluid flow is described by: d0 = c[v(/r.V0)] dt
(11)
where
3(1-2v)
a
ATS,
The total aqueous potential (pressure and chemical potential) of the pore fluid increases with the increase in pore pressure and/or chemical potential (decrease in salt concentration). When the total aqueous potential of the pore fluid increases, water will be adsorbed into the clay platelets. The water adsorption will result in either the platelets moving further apart i.e., swelling if they are free to expand, or the generation of hydrational stress if swelling is constrained. The process can be represented by (Tan et al., 1997): Ae^ =QiAlogO
(13)
where £« is strain due to change in rock water potential in direction i, Qi is volume change parameter at a given stress in direction i, and O is rock water potential. It is assumed that rock water potential will affect the rock matrix (effective) stress, and therefore the change in swelling strain can also be expressed as: Ae . =C.Aa.
(14)
where Q is volume change coefficient at a given stress in direction i, and a/ is effective stress in direction i. The volume change coefficient can be obtained from the following equation C; =
(15)
The generation of hydrational stress due to the rock material constrained from swelling is given by (12)
K a =l- — Ks where 5ij is Kronecker delta, K is drained bulk modulus of rock framework, and Kg is bulk modulus of mineral constituents. If a transient temperature field exists in the saturated rock, the difference in coefficient of thermal expansion of the pore fluid and the rock matrix can induce changes in pore pressure and stresses. This can lead to thermal-induced pore fluid flow and deformation of the rock around the borehole.
Ao^i=[D]^a^
(16)
where Goi is hydrational stress in direction i; and ID] is stiffness matrix.
3. INTEGRATION OF THE PROCESSES AND MECHANISMS The model takes into account changes in formation pore pressure arising from fluid flow and induced by deformation of the formation rock, and changes in effective stresses as a result of change in the formation pore pressure. In the mud pressure penetration model, the geometry of the pore network was represented as a series of parallel cylindrical tubes. Although the model
585 may not represent the actual flow processes in the pore network at microscopic scale, the approach enables certain properties of the porous medium (or equivalent continuum) for certain flow process at the macroscopic scale to be derived. The main effect of the mud pressure penetration process is in modifying the effective permeability. Chemical potential difference will affect the flow rate of dissolved solute ions and solvent (or pore fluid - the mud filtrate is considered to be pore fluid once it has infiltrated the pores of the shale). The main consideration of dissolved ions is their contribution to osmotic pressure. This is achieved by keeping track of the concentration of the dissolved ions (Equation 7). The effect of osmotic pressure on the flow of pore fluid is taken into account by including the osmotic pressure in the rock water potential, and use the spatial gradient of rock water potential (Equation 6) as the driving force for pore fluid flow. The effects of temperature can be prevalent for some formation rocks and in-situ conditions. Temperature-induced changes in pore pressure and rock matrix stress (thermo-poro-elasticity model) can be modelled using Equations 10 to 12. hi addition, to take into account the chemical potential effects, gradient of rock water potential instead of pore pressure gradient is used in Equation 11. As the effects of (rock matrix) swelling are mainly on effective stress, they can be taken into account by keeping track of changes in rock water potential or effective stresses, and the strain induced (swelling and hydrational stress model. Equations 14 to 16). The hydrational stress can be considered as additional internal stress in the equilibrium equations. For all the processes and mechanisms, the major coupling parameters are stress, pore fluid pressure and temperature.
dependent because some of the processes such as mud filtrate invasion, fluid flow and thermal diffusion are time-dependent. The results also show that thermoporoelastic ity effects can induce a pore pressure change of about 0.7 MPa close to the wellbore wall for every 1°C temperature difference between the drilling fluid and formation. The model results also show that the chemical potential mechanism can improve wellbore stability by lowering the pore pressure below the initial value in the formation near to the wellbore.
I 30 t-0 days t-0.01 days t-0.4 days t-2.5 days
1
1.5 2 Normalised distance from weHbore centre
2.5
Figure 1. The effects of coupled interaction of mechanical deformation, fluid flow & a colder drilling fluid on pore pressure.
4. MODEL RESULTS AND DISCUSSIONS Analyses have been conducted to study the effects of the different processes and their interaction on wellbore stability. These include coupling of some or all of the processes and mechanisms. Examples of the model results are shown in Figures 1 and 2. Figure 1 shows the change in pore pressure at different distance (normalized with respect to wellbore radius) from the wellbore center with time when drilling in a formation which is 30°C hotter than the drilling fluid, taking into account the effects of borehole deformation, mud filtrate invasion, thermal diffusion and thermoporoelastic effects. Figure 2 shows the model results which take into account borehole deformation, mud filtrate invasion, fluid flow, solute transport, chemical potential mechanism, and poroelastic effects. The model results show that stability can be time-
^lOdays ^•0.01 days • M).4days t-2.5 days
1
1.5 2 2.5 3 Nomfialised distancefromwellbore centre
Figure 2. The effects of coupled interaction of mechanical deformation, fluid flow & a lower water activity drilling fluid on pore pressure.
586 5. CONCLUSION A finite element program has been developed which can model the effects of the interaction of a number of important factors and processes on wellbore stability in shales. The factors and processes which have been incorporated into the model include the orientation of the wellbore, magnitude and orientation of the in-situ principal stresses, properties of the bedding planes, properties of drilling fluid and dissolved ions, mud pressure, formation pore pressure, temperature of drilling fluid and formation rock, and the fluid transport, thermal, geomechanical and properties of shale, swelling and hydrational stress mechanism, and thermoporoelastic effects. The results of some preliminary modelling studies showed that, as some of the important field variables such as formation pore pressure, rock aqueous potential, stress and temperature will change with time before a steady state condition is reached, wellbore stability in shales can be time-dependent and this must be taken into consideration in drilling fluid design and shale instability management. The models results in this paper show that the temperature or chemical potential of the drilling fluid can have significant influence on how the pore pressure field in the formation adjacent to the wellbore wall may change with time as a result of heat diffusion, thermoporoelastic effect, and chemical potential mechanism. Results of the modelling study conducted to date indicate that, in order to predict both short and long term stabilities of wellbore in shales, it is important to take into account the interaction of the processes and mechanisms described in this paper. The coupled model provides a tool for studying the relative importance of the various factors and processes on wellbore stability in shales, and how stability may change with time.
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