Saeed Rafieepour, Siavash Zamiran, Mehdi Ostadhassan

a University of Tulsa, Tulsa, OK, USA

b Southern Illinois University Carbondale, Carbondale, IL, USA

c University of North Dakota, Grand Forks, ND, USA

ABSTRACT Drilling through chemically-active shale formations is of special importance due to time-dependent drilling fluid—shale interactions. The physical models presented so far include sophisticated input parameters, requiring advanced experimental facilities, which are costly and in most cases unavailable. In this paper,sufficiently-accurate,yet highly practical,models are presented containing parameters easilyderived from well-known data sources. For ion diffusivity coefficient, the chemical potential was formulated based on the functionality of water activity to solute concentration for common solute species in field. The reflection coefficient and solute diffusion coefficient within shale membrane were predicted and compared with experimental measurements. For thermally-induced fluid flow, a model was utilized to predict thermo-osmosis coefficient based on the energy of hydrogen-bond that attained a reasonably-accurate estimation from petrophysical data, e.g. porosity, specific surface area (SSA), and cation exchange capacity (CEC). The coupled chemo-thermo-poroelastic governing equations were developed and solved using an implicit finite difference scheme. Mogi-Coulomb failure criterion was adopted for mud weight required to avoid compressive shear failure and a tensile cut-off failure index for mud weight required to prevent tensile fracturing. Results showed a close agreement between the suggested model and experimental data from pressure transmission tests. Results from a numerical example for a vertical wellbore indicated that failure in shale formations was time-dependent and a failure at wellbore wall after 85 min of mud—shale interactions was predicted. It was concluded that instability might not firstly occur at wellbore wall as most of the conventional elastic models predict;perhaps it occurs at other points inside the formation. The effect of the temperature gradient between wellbore and formation on limits of mud window confirmed that the upper limit was more sensitive to the temperature gradient than the lower limit.

Keywords:Chemo-thermo-poroelastic wellbore stability Shale—fluid interaction Chemo-osmosis Thermo-osmosis

1. Introduction

Maintaining stability of a wellbore and strengthening of the wellbore wall have been widely studied in the literature (Gholilou et al., 2017; Zhong et al., 2017, 2018; Gao and Gary, 2019; Singh et al., 2019a, b). Among these, wellbore instability in shales is of prominent importance as the shales are clay-rich formations forming the most abundant sedimentary rocks (around 70%). In addition to this,approximately 90%of drilling problems are related to wellbore integrity issues in shale formations (Rafieepour et al.,2015). These wellbore stability problems are attributed to the shale—fluid interactions that include various processes such as chemo-osmosis, thermo-osmosis, electro-osmosis, and swelling due to cation exchanges. Involving such time-delayed interactions in wellbore failure analysis refers to previous decades.However,oil industry has identified such interactions formerly and has taken advantage of oil-based drilling fluids to overcome related problems including packing off, hole enlargement, tight hole, etc. However,due to environmental issues regarding oil-based muds especially in offshore environments, the use of water-based muds is inevitable regardless of cumbersome drilling problems. The theory of irreversible or non-equilibrium thermodynamics is a convenient means of analyzing simultaneous flow of water, heat, and electricity in porous media. The literature is rich of the experimental and theoretical studies on the coupled processes in porous media. For example, Taylor and Cary (1960) reported that the experiments within thermal, electrical, and chemical gradients were impressed across saturated samples of a silt loam. Each of these gradients caused moisture movement through the soil that gave rise to other coupling effects.Katchalsky and Curran(1967)not only dealt with the fundamentals of the theory,but also suggested some practical applications to scientists and engineers. Gray (1966)experimentally investigate electrokinetics and thermal coupling in fully-saturated sodium clay-water systems. He also took advantage of irreversible thermodynamics as a theoretical framework for the investigation.

Coupling mutual effects of chemical potential and hydraulic pressure gradient into solute and solvent transport in shale membrane lead to decrease in shale strength due to near-wellbore pore pressure increase/effective stress decrease. Earlier works lumped the effects of solute transport and chemical potential gradient in the chemo-osmosis analysis. Chemical potential and membrane efficiencies were later incorporated into the coupled transport relationships (Sherwood, 1993; Yu, 2002). A complete chemical interaction based on non-equilibrium thermodynamics and mixture theory was presented by Sherwood (1993). A chemoporoelastic model was developed by Ghassemi and Diek (2002)in which the chemical potential of drilling fluid and formation fluid were described as a linear function of the mass fraction of solute.Ekbote and Abousleiman(2005)also proposed an analytical solution to a linearized anisotropic porochemoelastic model for inclined wellbores. Roshan and Rahman (2011a) proposed a fullycoupled chemo-poroelastic model for estimation of stress profile around the wellbore.

On the other hand, thermal-osmosis has been found to be an influence factor on fluid flow through soil systems. Derjaguin and Sidorenkov (1941) developed the physical principles of thermoosmosis at the molecular and pore scales. Regarding wellbore stability analysis,thermo-osmosis has not been taken into account as much as chemo-osmosis has been given(Nguyen and Abousleiman,2009; Roshan and Rahman, 2011b; Chen et al., 2015; Ostadhassan et al., 2015; Rafieepour et al., 2015a, b; Dokhani et al., 2016a).Moreover, in the studies that include thermal effects in the transport of volume fluids,there have been less regarding how thermoosmosis can be experimentally and theoretically estimated (or quantified).

In this study,a comprehensive model of transport phenomena was proposed involving shale—fluid interactions based on nonequilibrium thermodynamics. The governing equations were presented in cylindrical coordinate systems and solutions were assumed as non-ideal.Moreover,a model was used for estimation of thermo-osmosis coefficient based on work done by Goncalves et al. (2012). Several other models were used from the literature to estimate the reflection coefficient and the solute permeability in shale membranes. Then, the proposed model was compared with experimental results conducted by Ewy and Stankovich(2000). The effects of coupled processes on pore pressure and wellbore failure were also investigated. Based on the results,different wellbore failure types were observed including failure at borehole wall,failure inside the formation,and transient wellbore failure. Finally, the effect of the temperature gradient between wellbore and formation on the upper and lower limits of mud window was investigated. It should be noted that in this study, it was assumed that the medium was isotropic. In the literature,some studies have developed models for wellbore stability in shale formations considering the anisotropy effects (e.g.Ostadhassan et al.,2012;Zamiran et al.,2014,2018;Dokhani et al.,2016b).

2. Theory and model formulation

The phenomenological equation describes the linear relationship between the driving forcesXjand fluxesJj(Bader and Kooi,2005):

whereLijare the coupling coefficients that relate flows of typeito gradients of typej.The driving forces in fluid flow through a porous medium are usually pressure, salinity, electrical potential, and temperature gradients.In near equilibrium thermodynamics where the forces are small,the phenomenological equations are presented as (Jarzy′nska and Pietruszka,2008):

whereJv,JD,Iandqhare the molar total solution flux per unit pore cross-sectional area, molar diffusional flux of solute per unit pore cross-sectional area relative to the solvent flow, electrical current,and heat flux,respectively.Also,?P,?π,?Eand ?Tare the hydraulic potential gradient, osmotic pressure gradient, electrical potential gradient, and temperature gradient, respectively.Lijis the phenomenological coefficient which couples the fluxes with driving forces, with values being obtained from experiments.

A description of the direct and indirect (coupled) flows in shale formations and their corresponding driving forces is shown in Fig. 1. For near equilibrium systems, i.e. infinitesimal macroscopic gradients, the phenomenological coefficients can be considered to be constant and therefore, the transport equations are linearized. It is noteworthy that near equilibrium state, according to Onsager’s symmetry or reciprocal relations (ORR), in the range of equilibrium state, the cross coefficients must be equal to each other, i.e.Lij=Lji. In this study, it was assumed that the net electrical current was equal to zero. According to Rafieepour et al. (2015a, b), the Soret’s effect, Dufour effect,Seebeck effect, Peltier effect, electro-osmosis, streaming current electro-phoresis, and diffusion currents are negligible due to their small values.

In addition to this,several experimental observations confirm a violation from the Onsager’s reciprocity(Ghassemi and Diek,2002).It was assumed that the flow of solute and electrical current were not influenced by temperature gradients and the effects of hydraulic, chemical, and electrical potential gradients on heat flux were negligible.The heat flux was approximated by Fourier law of heat conductivity. It should be noted that due to the low permeability of shale membranes,a convective form of the heat transfer is negligible.

Under these assumptions, the coupled transport equations are as follows:

whereLPP,LPC,LPT,LCP,LCCandLTTare the coefficients of the membrane for filtration, chemo-osmotic, thermo-osmotic, ultrafiltration, ion diffusional, and heat transport, respectively. The overall solute flow through the membrane can be written as the following form:

Fig.1. Schematic diagram of coupling mechanisms for a fully coupled system in shales (Rafieepour et al., 2015a, b).

whereCsrepresents the bulk solute concentration inside the membrane system.

2.1. Chemical-osmosis

Osmotic pressure is the pressure required to prevent water from flowing through a semi-permeable membrane from a solution with low salt concentration to a solution with high salt concentration.Fig. 2 shows a semi-permeable clay membrane with a concentration gradient across it. The osmotic pressure can be presented as

Fig. 2. Illustration of osmotic pressure.

whereRis the universal gas constant,Tis the average shale temperature,awis the average activity of water,andVm，wis the average molar volume of water. The superscripts 1 and 2 stand for water activities of the two different solutions across the shale membrane.A shale membrane is ideal only in the presence of water-based muds containing low concentration of chemical agents to prevent the solute from diffusing into the formation and uniform small pore size distribution. This assumption should be justified for oil-based muds or water-based muds, with high concentration of chemical agents such as sodium silicates. Moreover, this is the case when a shale formation has a broad range of pore size distribution including large pore throats, which provide significant permeability to the solute constituent.Thus,chemical agents may not trap all the solute molecules and usually a few of these may diffuse into formation and influence fluid flow and consequently, change the pore pressure adjacent to the wellbore. The interactions of solute with the pore walls increase as the pore size is reduced that reduces the permeability of the membrane to solute.This phenomenon can be exerted into Eq.(5)by multiplying the right term of the equation by a coefficient called reflection coefficient (Im). This coefficient ranges from zero to one.For unconsolidated sandstones that freely allow flow of water and solute,the reflection coefficient is equal to zero and therefore,no osmotic flow can be found.For real shales,it varies between zero and one, and for the ideal membrane, it is equal to unity. Hence, the efficiency of clay membranes can be defined by

Water activity of an aqueous solution is a function of solute concentration, in mathematical form:

whereis the derivative ofawwith respect to the solute concentration.

From Eqs.(3)and(4)and after manipulation,the following can be derived:

where ω is the solute permeability coefficient and can be defined as

This coefficient represents the rate of solute diffusion across the membrane. When ω is zero, there is no diffusional flow (e.g. ideal membranes), and for non-selective membranes, we have ω =LCCCs. Therefore, solute flow in semi-permeable membranes for non-ideal solutions is given by

Because shale formations are composed of small plates of clay minerals and transport phenomena mostly take place close to the wellbore wall, here for simplicity, it is assumed that transport process is only inr-direction and thus, the continuity equation is simplified to the following relation:

where φ is the porosity of formation.

After substitution of Eq.(11)(by ignoring the last term)into Eq.(12), it can be reached to

Eq. (13) is the diffusion equation of solute through the porous medium. If solute transport via potential hydraulic gradient is neglected, then solute transport through the shales follows the Fick’s law of mass transfer by diffusion and becomes as

According to the above analysis, three phenomenological parameters are used to describe the behavior of non-ideal shale membrane systems (LPP,Im, ω). The hydraulic permeability coefficient or the mechanical filtration coefficient of a given membrane can be expressed as follows:

whereKis the hydraulic conductivity(e.g.in cm/s);andkand μ are the absolute permeability and viscosity,respectively.The reflection coefficient is representative of maximum expected osmoticallyinduced pressure. Marine and Fritz (1981) developed an equation for estimation of the reflection coefficient given as

whereKsis the distribution coefficient of solute within the membrane pores, which is the ratio of anion concentration within the membrane poresto the mean solute concentrationis the concentration of cations within the membrane pores as=The anion concentration can be obtained from Teorell-Meyer-Siever model:

whereCECis the cation exchange capacity of the clay(eq/g),ρ is the dry density (g/cm3), φ is the porosity of membrane, andis the mean concentration of solute on either side of the membrane(eq/cm3). For ideal membranes,is zero and thus the refelction coefficient is unity,while for non-perm selective porous media,it is equal toIf the value ofis too large, then the anion concentration approachesis the ratio of frictional coefficients between cation and anion with water in the membrane and is assumed to be 1.63.Rwmis the ratio of frictional coefficients between the cation and anion with the membrane structure and is assumed to be 0.1.Rmis the ratio of frictional coefficients between anion and solid membrane matrix to anion and water in the membrane structure and is considered to be 1.8. Detailed discussion is given in Fritz (1986). The last phenomenological parameter is the solute permeability coefficient. This parameter is a measure of the rate of solute diffusion from the side with high concentration to the side with low concentration.For ideal membranes,the solute can neither be transferred by advection (Im= 1) nor by diffusion(ω = 0).The value of ω depends onKs,frictional resistance within the membrane of anion with water (faw) and of anion with membrane structure (fam). The relation is expressed as

If the membrane is non-permselective (Im= 0) andKs= 1,which occurs when porosity is very high and thusRwm→0,Eq.(19)reduces to

2.2. Thermo-osmosis

Gray (1966) observed some pressure build-up by applying a temperature gradient across a clay sample which was indicated by the rise and drop of water level in standpipes on either side.Taylor and Cary (1960) outlined a theoretical analysis on the thermodynamics of irreversible processes to evaluate coupled flows of heat and water in continuous soil systems. The physical principles of thermo-osmosis in microscopic scales were proposed in 1941(Derjaguin and Sidorenkov, 1941). Although there is some advancement in the mentioned literature, there are still shortages for theoretical prediction of thermo-osmosis. Also, most of the theoretical models are in molecular and pore scale with too many parameters that are not easily measurable. Thus, it is required to upscale the microscopic thermo-osmosis coefficient to obtain macroscopic one for practical applications using ordinary and available data. Goncalves and Tremosa (2010) obtained the following equation for thermo-osmosis process based on the volume averaging method:

whereTis the absolute temperature (K), and ΔHis the fluid—solid interactions-induced macroscopic specific enthalpy change (J/m3).According to this study, flow of fluid occurs from the side with higher temperature to the side with lower temperature in the clay membrane (where ΔH>0). Goncalves and Tremosa (2010) developed a theoretical expression by directly formulizing the enthalpy changes due to hydrogen bonding modifications at the macroscopic scale. In the current study, the same model is used. Based on thermodynamics interpretation by Derjaguin and Sidorenkov(1941), the specific enthalpy is written as

where ΔHHBis the energy required to break one mole of hydrogen bonds; andCHBandare the concentrations of hudrogen bonds in the bulk system and pore fluid, respectively. The relations for these parameters can be given as follows:

where νw,ν+and ν-are the molar volumes of the water,cation,and anion, respectively. Using the Donnan equilibrium, the values ofC+andC-can be estimated as

whereeis the elementary charge,Nais the Avogadro’s constant,andQVis defined as the excess charge in a unit volume of porous medium per associated water volume based on following equation:

where ρsis the density of the solid.Mean half-pore size(b)can be derived from the following equation having specific surface area(SSA), considering a plane-parallel conceptual geometry for the porous medium:

By introducing Eq. (28) into Eq. (27), it is obtained thatQV= 96.3CEC/(Asb) = ξ/band ξ = 96.3CEC/As. In order to use this model for estimation of thermo-osmosis coefficient, it is only required to have some petrophysical properties along with some molecular parameters.For molecular properties,parameters in the literature can be used including= 3.5,V+= 23.8×10-6m3/mol, V-= 17.4×10-6m3/mol,Vw= 18×10-6m3/mol, ΔHHB= 14.5 kJ/mol,= 3.75, and a mean value of 1 nm forbs.

2.3. Volume flow through membrane

According to Eq. (2), the relationship for overall volume flow(solvent flow)is as follows:

From Eq. (12),LPC=-ImLPPand also from chemical potential gradient relation (?μ = (RT/Vm，w)[f′(Cs)/f(Cs)](dCs/dr)), the following relation for volume flow can be derived:

The continuity equation for flow of solvent through the porous media can be presented as

By introducing the Eq. (30) into Eq. (31) and by considering slightly compressible fluid,the diffusivity equation for solvent flow is as follows:

wherectis the total compressibility of the shale formation.

2.4. Heat flow in semipermeable membranes

As mentioned earlier, due to low permeability of shales, heat transport via convection is negligible and it is assumed that the conduction heat transfer is dominant which follows Fourier’s law:

wherekTis the thermal conductivity coefficient. In this study, it is also assumed that the heat transfer is only inr-direction and therefore, we have

This is the heat diffusivity equation andChis the thermal diffusivity coefficient given byCh=kT/(ρcp).

2.5. Boundary and initial conditions

The geometry of borehole stability problem is presented in Fig. 3. To determine the solute concentration, pressure, and temperature profiles, nine initial and boundary conditions must be specified. These conditions for the model can be summarized as follows:

3. Stress distribution around wellbore

The general solution to the thermo-poro-elastic wellbore stability model is determined via the superposition principle by combining the mechanical, hydraulic, chemical, and thermal induced effects. The stress—strain relations for a chemo-porothermoelastic medium are written as

where σij,εijand εkkare the total stress,total strain,and volumetric strain, respectively; and α,G, ν, αmand δijare the Biot-Willis effective stress coefficient,shear modulus, Poisson’s ratio, volumetric thermal expansion coefficient, and the Kronecker delta,respectively. In Eq. (36), compressive stress is assumed to be positive. The complete form of the stress components around the wellbore including thermo-poroelastic effects is expressed as follows (Li,1998; Yu, 2002):

Fig. 3. (a) Schematic diagram for a wellbore subjected to anisotropic in situ stress field; and (b) Problem domains.

whererwis the wellbore radius and θ is the angle around the wellbore.

4. Failure criterion

The Mohr-Coulomb (M-C) failure criterion is widely used in the area of borehole instability analysis. However, this criterion ignores the effect of intermediate stress component. Al-Ajmi(2006) introduced a poly-axial failure criterion that considers the effects of intermediate principal stress in the shear failure analysis. This failure envelope has been used in this study as defined by

where φ is the internal friction angle; σm，2is the effective normal stress and τoctis the octahedral shear stress defined by

where σ1, σ2and σ3are the major, intermediate and minor principal stresses, respectively.

A failure index (FI) is usually defined to evaluate the wellbore conditions in terms of collapse and tensile failures.For any type of failure to occur,FImust become negative. For the case of Mogi-Coulomb failure criterion, the collapseFIis given as

This failure function was successfully performed in a field study in offshore Iran for well path optimization(Rafieepour and Jalalifar,2014). Moreover, a tensile fracturing is probable when the minimum effective principal stressin the rock formation surpasses the tensile strength (McLean and Addis, 1990). The breakdownFIis given by

whereT0is the tensile strength of the rock.

5. Numerical method and computer implementation

Solute concentration, pressure, temperature, and stress distributions are the four primary unknowns in the proposed model,which shall be determined to evaluate the wellbore stability. The system of equations presented is nonlinear due to the dependency of the coefficients to the mentioned unknowns. Consequently, a numerical approach shall be considered to solve for these unknowns under specific initial and boundary conditions. There are advanced numerical schemes, e.g. finite element method, which are suitable for solution of complex geomechanical problems(Zhai et al., 2009). However, the finite element method is a timeconsuming and computationally expensive approach. For these reasons, the system of equations is solved using an implicit finite difference scheme. The constant time step and mesh size finite difference approach is adopted to solve the proposed initialboundary value problem. The flowchart for wellbore stability analysis is summarized in Fig. 4.

6. Results and discussion

Fig. 4. Flowchart of wellbore stability model.

As mentioned previously, estimation of thermo-osmosis coefficient for wellbore stability analysis in shale formations has not been taken into account. The previous studies simply have presented assumed amounts for this coefficient in the analyses without any discussion on how this parameter can be estimated.In this investigation,the proposed model is validated for prediction of thermo-osmosis coefficient through comparison with laboratory works conducted by Gray (1966) on saturated clay-water electrolyte systems.Gray(1966)performed various experiments on a pure kaolinite.Based on Gray(1966),the sample was firstly washed in a concentrated solution of sodium chloride,and after drying at 230°F(110°C), it was mixed with 0.001 mol/L NaCl solution.Subsequently, the clay sample was placed in a flow cell with pressure, thermal, and electrical gradients applied on both ends.The temperature gradient across the clay sample was established by internal heating and cooling coils. The temperatures were monitored using a digital thermometer. The resulting changes in pressure were indicated by the rise and fall of water level in standpipes on both sides of the flow cell.The temperature-induced pressure was observed to be directly proportional to the difference in water level in the two pipes. By applying a thermal gradient of 1.08°C/m and a mean temperature of 26.8°C at the steady state condition, the difference in water level in the two standpipes was measured as 0.51 cm/°C. SSA value was estimated as 50 m2/g.Consequently, the mean half pore size would be 4.094 nm. Using the data shown in Table 1 and applying the thermo-osmosis coefficient model,the estimated value for ΔHis equal to - 15.1 kJ/m3.Substitution of specific enthalpy, mean temperature, permeability,and viscosity into Eq.(21)results in Δh/ΔT= 0.52 cm/°Cwhich is 0.06 cm/°Chigher than measured value for temperature-induced pressure (about 12% error). Then, the thermo-osmosis coefficient is equal toLPT= - 2.01× 10-10. The negative value for the thermo-osmosis coefficient reveals that the flow occurs from the cold side to the hot side. The above analysis shows that the theoretical model based on the alteration in hydrogen bond network of water molecules can admittedly predict the results of experimental work conducted by Gray(1966).

The above analysis focused on estimation of the thermoosmosis coupling coefficient. The relations presented in section for chemo-osmosis and several pressure transmission tests conducted by Ewy and Stankovich (2000) are utilized to estimate the reflection coefficient and permeability coefficient of solute species.Ewy and Stankovich (2000) performed a series of tests on shale samples under simulated in situ conditions. They developed a technique for measuring changes in shale pore pressure caused by simultaneous application of hydraulic and osmotic gradients.They used three shale samples in their experiments(A1,A2 and N1).Only shale samples of A2 and N1 showed significant membrane behavior. The pore pressure at the outlet end was measured and recorded continuously. The data used for shale sample N1 contacting CaCl2solution are tabulated in Table 2.Based on these data,values of 0.068 and 5.3×10-13mol/(Pa m s)were obtained for the reflection and solute permeability coefficients, respectively. Ewy and Stankovich (2000) estimated a reflection coefficient of 0.02 by contacting shale sample N1 contacting a 267 g/L CaCl2solution and plotting the osmotic pressure versus the fluid activity difference. Results show good agreement between modeling and experimental data.

To understand the effect of chemo-osmosis (under isothermal conditions) on the pore pressure with time forT0=Tw，P0= 15 psi， andPw= 985 psi (1 psi = 6895 Pa), the correlation between water activity and solute concentration was used, as shown in Fig. 5. Wellbore geometrical data used here are from Table 3.Fig.6 indicates the transient pore pressure profile for various exposure times. The pressure inside the formation increases with time and sweeps farther distances into the formation.However, after 15 h of exposure, the profile of pore pressure is stable and has not been changed with time.This implies a balanced condition of chemical activities of formation water and drilling fluid in all points inside the formation. In the next step, a wellbore stability analysis was performed based on the general coupled model presented in previous sections.

Table 1Parameters for kaolinite shale reported by Gray(1966).

Table 2Parameters for shale N1, reported by Ewy and Stankovich (2000) and estimated based on Simpson (1997).

6.1. Wellbore failure modes

As mentioned before,wellbore failure occurs whenFI＜0.TheFIis a function of time and space and strongly depends on the state of effective stress in the rock. Three different scenarios are probable for compressive failure around a wellbore: transient (time-dependent) borehole failure, failure occurring at the wellbore wall, and failure occurring at some distance from the wellbore wall (i.e. inside the formation).

6.1.1. Transient compressive failure

Effects of thermo- and chemo-osmosis for the case of heating(negative temperature gradient) andCdf＜C0are shown in Fig. 7a and b.The data used in the simulations can be found in Tables 3 and 4 and the correlation of water activity and solute concentration in Fig. 5. As it can be seen, fluid flow due to hydraulic pressure gradient is lower than any other flow mechanisms. According to Fig. 7a, the combination of thermo- and chemo-osmosis including hydraulic potential increases pore pressure inside the formation and triggers instability of the formation around the wellbore(peak pore pressure of 23.5 MPa). This can also be observed as large negative values of collapse failure index (collapse inside the formation) in Fig. 7b. In this example, the effect of chemo-osmosis is more visible than that of thermo-osmosis(large chemical potential and small temperature gradients).

Fig. 5. Water activity (aw) versus solute concentration for CaCl2.

Table 3Rock and fluid properties,well and in situ stress data(Ewy and Stankovich,2000;Yu,2002; Rafieepour et al., 2015a, b).

Fig. 8a—c shows the contour diagrams of collapse failure index values around a vertical wellbore over elapsed time.This case was designed such that a compressive failure is probable at the wellbore wall due to using a drilling fluid with low density.As it can be seen,the wellbore is stable initially but shortly after shale-drilling fluid exposure (transport of fluid into the formation due to hydraulic transport,chemo-and thermo-osmosis changes pore pressure),an unstable region is formed,and wellbore wall fails after 85 min.It is also clear from these figures that failure around the wellbore occurs along the direction of the minimum horizontal stress, due to the horizontal stress anisotropy in the ground.

Fig.6. Pore pressure distribution of shale in contact with CaCl2 solution for T0 = Tw，P0 = 15 psi， and Pw = 985 psi.

Fig. 7. Effect of thermo- and chemo-osmosis on the pressure profile for the case of heating and Cdf ＜C0: (a) Plot of pore pressure; and (b) Failure index.

6.1.2. Compressive failure at wellbore wall

An example of wellbore stability analysis for a vertical well with anisotropic horizontal stresses is presented in Fig. 9. In this case,due to low mud weight, wellbore failure begins at the wall.Moreover, due to horizontal stress anisotropy, the failure at wellbore wall is not uniform(symmetric)and the breakout extension is along the minimum horizontal stress component.Also,from Fig.9,theFIat outer boundary is equal to 7 MPa while its value in the inner boundary (wellbore wall) is -0.5 MPa. In this case, the hydrostatic pressure of mud column is responsible for failure at wellbore wall. The most suitable and effective remedial act in this scenario is raising mud weight.

6.1.3. Compressive failure inside the formation

Another case that can be considered is that a vertical wellbore is drilled through a chemical active shale formation with a large chemical potential/temperature gradient between wellbore andformation, i.e. negative temperature (formation heating) and positive concentration/negative water activitygradients.In such circumstances,fluid transport into the formation occurs under various mechanisms including the direct flow of volume fluid due to hydraulic potential gradient, and indirect flow due to chemoand thermo-osmosis gradients.The combination of these transport processes causes a significant rise in pore pressure around the wellbore.Fig.10a shows pore pressure profile for the case of ΔT=Tw-T0= 49.45°C and= -3.9 mol/L.Change in pore pressure profile re-distributes stress concentration around the wellbore. Stress concentrations exceeding rock strength form a failure region inside the formation are shown in Fig. 10b, with negative failure index values of about -3 MPa.

Table 4Hydraulic, chemical and thermal properties (Ewy and Stankovich, 2000; Yu, 2002;Rafieepour et al., 2015a, b).

6.2. Effect of temperature gradient on critical mud weight

Fig.11 shows the effect of the temperature difference between wellbore and formation on the critical mud weights required for prevention of borehole fracturing and collapse. As it can be seen,increasing temperature increases both upper and lower limits of mud window. However, the effect of temperature gradient on critical breakdown pressure is higher than that of breakout mud weight.For example,formation heating by 10°C causes increasing in fracturing mud pressure by 1.6 MPa (a fracturing mud weight increase of 0.35 ppg,1 ppg = 0.001176 MPa/m) while it increases collapse pressure by 0.35 MPa (a collapse mud weight increase of 0.0725 ppg).

7. Conclusions

In this paper, several models were presented for prediction of chemo- and thermo-osmosis parameters from well-known data sources. For ion diffusivity coefficient, the chemical potential was formulated based on the functionality of water activity to solute concentration for common solute species in field. For thermallyinduced fluid flow, a model was utilized to predict thermoosmosis coefficient based on the energy of hydrogen-bond that attained a reasonably-accurate estimation from petrophysical data,e.g. porosity, SSA, and CEC. A coupled chemo-thermo-poroelastic model was presented and the governing equations were solved using an implicit finite difference scheme. Results confirmed that chemical and thermal effects significantly influenced formation fluid content and displacement field near the wellbore wall.It was found that various types of wellbore failure may occur around the wellbore including failure at wellbore wall, failure inside the formation, and time-dependent failure. Wellbore failure at wall occurred when the hydrostatic effects were dominant with no transport of fluid into/out of formation. This is what most of the conventional elastic models predict.

Fig. 8. Contour diagrams of failure index around the wellbore: (a) After 20 min; (b)After 30 min; and (c) After 85 min and the wellbore failure begins.

Fig. 9. Failure at wellbore wall.

Fig.10. (a) Pore pressure profile around the wellbore; and (b) Failure at some points inside formation.

Fig. 11. Critical mud weight with various temperature differences between wellbore and formation.

In this study,it was demonstrated that in chemically active shale formation and when there was a temperature gradient, wellbore failure may occur firstly at some points inside the formation rather than at wellbore wall, i.e. for the case of formation heating and negativewater activitygradient.Moreover,the effectof temperature gradienton criticalcollapse and fracturing mud weights was investigated. Findings showed that heating the formation increased both mud limits; however, the breakdown mud weight was influenced more by temperature rise than breakout mud weight.For wellbore instability problems in shales,there have been several experimental studies which show the effectiveness of nano-particles on wellbore stability enhancement in shale formations, based on the results from triaxial strength measurements(Gao et al., 2016). These experimental investigations can be considered for further study in this area.

Declaration of Competing Interest

The authors wish to confirm that there are no known conflicts of interests associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Acknowledgments

The authors would like to acknowledge the financial and technical supports from the Petroleum Engineering Department at the University of North Dakota.