1 INTRODUCTION

Shale formations play host to approximately 75% of all drilled sections in the petroleum indus-try and, largely due to detrimental interactions between drilling fluids and the formation, ac-count for about 90% of wellbore-stability-related drilling problems (Steiger & Leung 1992). Shale formations are problematic because salinity (or chemical potential) differences between the wellbore fluid and formation can drive shale swelling and pore-pressure increase in the for-mation, leading to wellbore instability. However, adding salt to the drilling fluid can halt or even reverse the direction of this osmotic flow, leading to shrinkage and pore pressure decrease in the borehole wall and, therefore, a temporary stabilizing effect until the casing can be in-stalled.

In the current state of the industry, the design of drilling fluid chemistry is often based on characterizing the parameters that determine the response of a shale to a change in the salinity of wellbore fluid. The mechanical wellbore-stability is rarely considered in the design. Several researchers have contributed to extend the theory of poro-elasticity pioneered by Biot to include physico-chemical processes (e.g. Sherwood 1994, Heidug & Wong 1996, Coussy 2004, Sarout & Detournay 2011), and analytical solutions to chemo-poro-elastic problems involving simple geometries have been published in the literature (e.g. Ekbote & Abousleiman 2003, Bunger 2010, Sarout & Detournay 2011). However, in spite of a number of various approaches to nu-

Modeling the impact of drilling fluid salinity on wellbore stability using FLAC

C. Detournay Itasca Consulting Group, Inc., Minneapolis, MN, USA

A. P. Bunger CSIRO Earth Science and Resource Engineering, Melbourne, Australia Dept of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania, USA

B. Wu CSIRO Earth Science and Resource Engineering, Melbourne, Australia

ABSTRACT: A numerical implementation of chemo-poro-elastoplastic governing equations is proposed in FLAC for the purpose of predicting the impact of drilling fluid salinity on the sta-bility of petroleum wells that are drilled through clay bearing, or so-called “chemically active” rocks such as shale. The implementation takes advantage of the observation that, with osmotic pressure interpreted as temperature, similarities can be drawn between the governing equations for chemo-poro- and thermo-poro-mechanics. The main differences are the transport equations, which are coupled in the chemo-poro-mechanic logic (versus uncoupled in thermo-poro-mechanics). The logic is applied to an elastic oedomer test case for which an analytical solu-tion exists, and comparison of numerical and analytical solutions for the verification test show an excellent match and demonstrating that the impact of volumetric strain and pore pressure on osmotic pressure is of second order for this configuration.

A parametric analysis is then carried out for a borehole case study, with a Mohr-Coulomb constitutive model adopted for the shale. The analysis shows that an increase in reflection coef-ficient promotes borehole stability, and that the parameters that couple the volumetric strain and pore pressure with the osmotic pressure have little effect on evolution of the pore and osmotic pressure for the conditions considered in this study. In fact, for the borehole case, it is the con-sideration of cross-flow in the transport equations, that is, the transport of the fluid phase coun-ter to the osmotic pressure gradient, that accounts for the leading mechanism involved in the coupled process.

merically modeling the chemo-hydro-mechanical coupling in this problem, plasticity is usually only inferred based on an analysis of the elastic stress field (Ghassemi & Diek, 2003, Ghassemi et al. 2009, Roshan & Rahman 2011). Only recently have Roshan & Fahad (2012) presented evolution of a plastic zone for a chemo-poro-mechanical model that uses a Cam Clay plasticity criterion. But, in general, tracking the development of the plastic zone in these coupled systems remains a numerical challenge that has not been widely addressed and hence the vital issue of plastic deformation around the wellbore in a chemo-poro-mechanical framework remains unre-solved.

This paper introduces a numerical implementation of the chemo-poro-elastoplastic governing equations in FLAC (Itasca 2011) for use in studying wellbore stability in shale formations. Nu-merical pore pressure predictions for an oedometer test are compared to analytical solutions. A generic example is used to study chemo-poro-mechanical effects on wellbore stability, and sen-sitivity analyses are carried out to evaluate the impact of the reflection coefficient, and of the parameters characterizing the physico-chemical interaction between pore fluid and solid matrix on yielding damage around the wellbore.

2 THEORETICAL FRAMEWORK

The governing chemo-poro-elastic equations in this paper are based on the theoretical develop-ment of Sarout & Detournay (2011) and Bunger (2010). The formulation uses pore pressure, osmotic pressure and effective stress as main variables. Also, by convention stress/strain is tak-en positive in tension/extension. The laws describing the physics of the chemo-poro-elastic pro-cess are as follows.

2.1 Constitutive Laws (volumetric form)

1

bp bCC

(1)

2

2

1p b b C S

S b C

(2)

2

2 2

1b b C S p

b C

(3)

where is mean stress, p is pore pressure, is osmotic pressure, is volumetric strain, is the variation of fluid content, is the relative increment of salt content, C is compliance (the in-verse of drained bulk modulus 1/K), b is Biot’s coefficient, S is the unconstrained storage coef-ficient (S = bC/B, where B is the Skempton pore-pressure coefficient), and , , are parame-ters characterizing the physico-chemical interaction between pore fluid and solid matrix (, are dimensionless; has dimension stress

-1.) The impact of a change in both the volumetric

strain and the pore pressure on the osmotic pressure are assumed to be negligible in the numeri-cal development documented in this paper; accordingly, the following reduced form for Eq. (3) is used:

2 2

1

b C

(4)

2.2 Transport Laws

c

q p R

Dr R p

N

(5)

where q is the specific discharge, r is the relative solute flux, is the mobility coefficient, R is the reflection coefficient, Dc is the ion diffusion coefficient, and N is a (constant) parameter with dimension of stress.

2.3 Mass Balance

q

r

(6)

The chemo-poro-elastic equations can be compared to the thermo-poro-elastic equations im-plemented in the FLAC thermo-poro-mechanical logic.

3 NUMERICAL IMPLEMENTATION

The numerical implementation relies on similarities in the equations describing the chemo-poro-mechanical process and the thermo-poro-mechanical process, which is already an availa-ble feature in FLAC. With osmotic pressure being interpreted as temperature, the set of consti-tutive equations have similar functional form. An important difference concerns the transport Equations (5), which are coupled in the chemo-poro-mechanical logic, versus uncoupled in thermo-poro-mechanics. In fact, it will be shown in a borehole stability case study, that it is the consideration of cross-flow in the transport equations that accounts for the leading mechanism involved in the coupled chemo-poro-mechanical process.

3.1 Numerical approach

The chemo-poro-elastic logic documented above is extended to account for elasto-plastic mate-rial behavior. The extended (incremental) equations are implemented in the numerical code FLAC by taking advantage of the thermal logic already available in the code, and the parallel between this, and the chemo-poro-elasto-plastic logic. As mentioned above, it is assumed that the impact of volumetric strain and pore pressure on osmotic pressure can be ignored — i.e. the last two terms in Eq. (3) are neglected. The implementation holds for plane-strain conditions, and is done using the FISH built-in code language. The logic can be used with any of the consti-tutive models available with FLAC.

For the osmotic logic, initial conditions are given in terms of osmotic pressure; boundary conditions are either a given solute flux (zero by default) or a given osmotic pressure. The cross terms in the transport equation are accounted for in the FLAC logic by incrementing the nodal flow/flux by a corresponding contribution using FISH access developed for this purpose. Fluid, ‘osmotic’ and mechanical simulations are carried out in sequence. Mechanical readjustments to fluid and osmotic pressure changes are assumed to take place instantaneously in the simula-tions. In contrast, fluid and osmotic processes evolve with their own time scales, governed by their respective diffusivities. One important aspect of the coupled simulations is the synchroni-zation of fluid and ‘osmotic’ times that is being enforced using a FISH procedure.

4 VERIFICATION TEST

An elastic oedometer test is simulated with FLAC using the new logic, and a comparison of re-sults is made with the analytical solution presented by Sarout & Detournay (2011). The problem is one-dimensional. The height of the model is taken as 1 m for the verification example. The x-

reference axis is along the bottom of the model, and the y-axis is pointing up. The property val-ues for the test are listed in Table 1.

Fluid pore pressure and osmotic pressure are zero initially in the model. The bottom and sides of the oedometer box are impermeable to both solute and solvent and the mechanical boundary conditions correspond to roller boundaries. Fluid pressure is fixed at zero at the top of the box, and vertical displacement is free. Plane-strain conditions apply. Osmotic pressure is raised instantaneously to 0 = 1 Pa at the top of the model, which is obviously smaller than the order of MPa increase used in laboratory experiments such as those of Sarout and Detournay 2011, however it is convenient for the purpose of benchmarking. The simulation is carried out for a total of 1 × 10

12 sec, with intermediate results at 4.2 × 10

10 and 16.8 × 10

10 sec, where again

we note this is a much longer timescale due to the 1 m long specimens than would ever be en-countered in practical applications. The pore and osmotic pressures at elevation 0 m, 0.5 m, and 0.8 m from the bottom of the box are monitored during the simulation. The FLAC grid contains a total of 20 zones; it is shown in Figure 1. The history of pore pressure at monitoring points is plotted versus flow time for up to 4.2 × 10

10 sec in Figure 2.

Pore pressure is seen to decrease in the model as osmotic pressure diffuses into the box. The maximum magnitude of pore-pressure drop, scaled by imposed osmotic pressure, is about 0.15. The number is consistent with the value selected for reflection coefficient, R. Also, beyond the peak, pore pressure evolves back to the initial value for large-enough simulation times, as shown in Figure 3.

Table 1. Properties for the oedometer test simulation. ________________________________________________________________________________________________________

Property Value Units ________________________________________________________________________________________________________

Compliance, C 7.7 10-10

Pa-1

Poisson ratio, 0.4 -- Skempton coefficient, B 0.95 -- Biot coefficient, b 0.95 -- Reflection coefficient , R 0.15 -- Mobility coefficient, 3 10

-19 m

2Pa

-1s

-1

Ion mobility coefficient, Dc/N 1 10-19

m2Pa

-1s

-1

Alpha, 0.1 -- Beta, 0.25 -- Gamma, 1 10

-8 Pa

-1

Porosity, n 0.1 -- Material dry density, 1000 kg.m

-3

________________________________________________________________________________________________________

Figure 1. FLAC grid for the oedometer test.

Figure 2. Pore pressure versus time at monitoring points in the oedometer test for up to 4.2 × 10

10 sec.

Figure 3. Pore pressure versus time at monitoring points in the oedometer test for up to 16.8 × 10

10 sec.

Evolution in time of osmotic pressure at monitoring points is shown for up to 4.2 × 10

10 sec

in Figure 4. The osmotic pressure front is seen to propagate downward into the model. When the simulation is continued further, up to 16.8 × 10

10 sec, a uniform value of 1 Pa is reached in

the model as shown in Figure 5. Vertical displacement at monitoring points is plotted versus flow time for up to 4.2 × 10

10 sec

in Figure 6. The model is shown to settle and then heave as time evolves toward steady-state. (Note that the maximum magnitude of displayed displacements is small compared to 1. Howev-er, the trends are meaningful: in this elastic case, pore pressures and displacements scale by the imposed value of osmotic value.) The displacement results for up to 16.8 × 10

10 sec are shown

in Figure 7. The long-term surface displacement (top of model), u

y , can be evaluated as follows. For the

oedometer conditions, x = z = 0, and the stress-strain relation in the y-direction simplifies to (see Eq. (1))

3 1 1 1

1 3 1yy yybp bC

C

(7)

Also, by mechanical equilibrium, yy = 0, and, in the long term, p = 0 and = 0; thus, Eq. (7) gives

0

1

3 1y

Hu bC

(8)

where H is the height of the box (1 m for the simulation).

With the properties and osmotic pressure used for the simulations, u

y is 5.689 × 10

-11 m, and

the surface displacement predicted by the numerical simulation at long time (t = 1012

sec) is 5.405 × 10

-11 m. The relative discrepancy is 5%.

Comparison of pore pressure predictions with the analytic solution for this problem (Sarout & Detournay 2011) is shown in Figure 8.

Figure 4. Osmotic pressure versus time at monitoring points in the oedometer test for up to 4.2 × 10

10 sec.

Figure 5. Osmotic pressure versus time at monitoring points in the oedometer test for up to 16.8 × 10

10 sec.

Figure 6. Vertical displacement versus time at monitoring points in the oedometer test for up to 4.2 × 10

10

sec.

Figure 7. Vertical displacement versus time at monitoring points in the oedometer test for up to 16.8 × 10

10 sec.

Figure 8. Comparison of pore pressure predictions with the analytical solution at monitoring point in the oedometer test for up to 4.2 × 10

10 sec.

The results in Figure 8 show that the match between numerical and analytical solutions is

very good; in fact, the two solutions cannot be distinguished at the scale of the plot. Also, the assumption (adopted in this work) of negligible impact of volumetric strain and pore pressure on osmotic pressure (see last two terms in Eq. 2) appears to be justified, at least for the case in-vestigated.

5 PARAMETRIC STUDY

The parametric study of a vertical wellbore in a Mohr-Coulomb rock mass is considered in this analysis. The radius of the borehole is 0.155 m. The plane of analysis (normal to the borehole axis) is located at 3000 m depth. The stress state is anisotropic: the vertical stress at 3000 m depth is 67.9 MPa, the minimum and maximum horizontal stresses are 47.5 MPa, and 57.5 MPa, respectively. The initial pore pressure is 30.5 MPa. Mud pressure is 40.5 MPa, and osmot-ic pressure is 20 MPa.

The parametric study looks at the effect of parameters and , and the reflection coefficient, R on the distribution of pore and osmotic pressure and the extent of plastic yielding in the mod-el under consideration. The cases investigated in the parametric study are listed in Table 2. The value of gamma is 1 × 10

-8 in all cases treated. Note that the parameters and are taken to be

equal in the parametric study.

The grid used for the FLAC simulations is shown in Figure 9; it takes advantage of quarter symmetry and contains a total of 30 × 30 zones. The outside boundaries are located 10 radii away from the borehole center. The initial horizontal stress is 47.5 MPa in the x-direction, and 57.5 MPa in the y-direction; the initial out of plane stress is 67.9 MPa. The mechanical bounda-ry conditions correspond to roller boundaries along the lines of symmetry; displacements are fixed along the outside boundaries. Fluid and osmotic flow are not allowed to take place across the model outside boundaries, which are impermeable.

Table 2. Parametric study for the wellbore problem. ________________________________________________________________________________________________________

Case 1 2 3 4 5 6 7 8 ________________________________________________________________________________________________________

alpha=beta 1.0 1.0 1.0 1.0 0.0 0.0 0.0 0.0 R 0.0 0.1 0.3 0.5 0.0 0.1 0.3 0.5 ________________________________________________________________________________________________________

Figure 9. FLAC grid for the borehole examples.

Fluid and Mohr-Coulomb properties for the rock are the same as those adopted for the

oedometer example (see 0) except for the reflection coefficient, and parameters and which take the values listed in Table 2.

The hole is excavated instantaneously in the simulation, and a mud pressure of 40.5 MPa (overbalanced) is applied at the wall immediately after excavation. It is assumed that pore fluid in shale and mud have similar properties and there is not a mud cake on the borehole surface.

The simulations are carried out in two steps. In the first step (step 1, short-term response), the hole is excavated, mud pressure is applied at the wall, and the undrained response is calculated. No fluid or osmotic flow is occurring in this step. In the second step (step 2), flow is allowed to take place, and mechanical readjustments are computed. The transient coupled simulation is carried out for a total of 1 × 10

8 sec in all cases considered in the analysis.

5.1 Simulation Results for Step 1

The undrained response is the same for all simulation cases. (Osmotic pressure has no impact on model response at this stage.) The magnitude of imposed mud pressure (40.5 MPa) is less than the magnitude of the minimum in-plane in-situ stress (47.5 MPa); thus, a certain amount of hole closure is expected to take place. Displacement vectors in the vicinity of the hole at the end of simulation step 1 are shown in Figure 10.

The predicted hole closure (i.e. twice the magnitude of maximum radial displacement shown in Fig. 10) is about 1.9 mm in the x-direction, and 7.2 mm in the y-direction. Volumetric defor-mations induce pore pressure changes in the formation. A plot of pore pressure contours in the vicinity of the hole at the end of step 1 is shown in Figure 11. The pore pressure near the bore-

hole is seen to decrease in the y-direction (max in-situ stress), and to increase in the x-direction (minimum in-situ stress), as expected. Also, results show that at the end of step 1, limited yield-ing has taken place in a near-surface region of the formation, see Figure 12, which is not un-common observation in field situations.

Figure 10. Displacement vectors in the vicinity of the hole, undrained response.

Figure 11. Pore pressure contours in the vicinity of the hole, undrained response.

Figure 12. Indicator of plastic yielding in the vicinity of the hole, undrained response.

5.2 Simulation Results for Step 2

The effect of the reflection coefficient on borehole stability is considered first in this section, the effect of parameter ( = ) is analyzed next.

5.2.1 Effect of reflection coefficient for given = As fluid flows into the formation, pore pressure rises in the rock mass. Material yielding is ex-pected to develop if pore pressure increases to a high-enough level. This situation is considered in the simulation with strong coupling between the osmotic pressure and the volumetric strain/pore pressure but without cross-flow (Table 2, case 1). Contours of pore pressure and yielding zones at the end of the simulation case 1 are shown in Figure 13.

The simulation is repeated, with a reflection coefficient of 0.5 (case 4). The coupled chemo-poro-mechanic simulation is carried out for a total simulation time of 1 × 10

8 sec. Contours of

pore pressure and extent of material yielding at the end of the simulation is shown in Figure 14.

Figure 13. Pore pressure contours (left) and yielding zones (right) in hole vicinity, t = 1×10

8 sec – case 1.

Figure 14. Pore pressure contours (left) and yielding zones (right) with cross flow, t = 1×10

8 sec – case 4.

As osmotic flow takes place into the formation for this case, a cross-flow of formation fluid

tends to develop toward the hole in an attempt to equilibrate salt concentration. As a conse-quence of this mechanism, the magnitude of pore pressure rise in the formation is seen to be smaller than in the simulation case without cross-flow. Also, the extent of the yielding region is smaller, as expected.

Contours of osmotic pressure at t = 1 × 108 sec for simulation case 1 and 4 are shown in Fig-

ure 15. In Figure 16, pore pressure at the end of the simulation is plotted versus radius along the symmetry lines (x- and y-directions) for different values of the reflection coefficient (cases 1 to 4.)

The results in Figure 16 clearly show that higher pore-pressure reduction is predicted to oc-cur for larger value of the reflection coefficient in the simulations. Osmotic pressure profiles at the end of the simulation are shown in Figure 17.

Figure 15. Osmotic pressure contours at t = 5 × 10

9 sec for case 1 (left) and case 4 (right).

Figure 16. Pore pressure at t = 1×10

8 sec versus radius in x-direction (left) and y-direction (right) for =

= 1 and R = 0.0, 0.1, 0.3 and 0.5.

Figure 17. Osmotic pressure at t = 1 × 10

8 sec versus radius in x-direction (left) and y-direction (right) for

= = 1 and R = 0.0, 0.1, 0.3 and 0.5.

The increase in reflection coefficient tends to (slightly) reduce osmotic pressure diffusion in-

to the formation, hence less salt is introduced in the formation. The decrease in pore and osmot-ic pressures in the formation associated with an increase in the reflection coefficient have a pos-itive impact on wellbore stability [we are assuming in the context of this paper that shale strength is not altered by salt].

5.2.2 Effect of coupling coefficients = for given reflection coefficient. The simulation cases 1-4 are repeated, using a zero value for = . Pressure and osmotic pro-files for cases 5 to 8 are compared to those obtained for cases 1to 4 in Figures 18 & 19.

As the results in Figure 18 & 19 show: for fixed value of the reflection coefficient, a varia-tion of = between bounding values of zero and 1 has little effect on the pressure solution in the particular problem conditions considered in the study.

Figure 18. Pore pressure at t = 1 × 10

8 sec versus radius in x-direction (left) and y-direction (right) for =

= 0 (line) and = = 1 (symbol) at R = 0.0, 0.1, 0.3 and 0.5.

Figure 19. Osmotic pressure at t = 1 × 10

8 sec versus radius in x-direction (left) and y-direction (right) for

= = 0 (line) and = = 1 (symbol) at R = 0.0, 0.1, 0.3 and 0.5.

6 CONCLUSIONS

A numerical implementation of the chemo-poro-elastoplastic governing equations in FLAC is used in this study. The implementation takes advantage of the observation that, with osmotic pressure interpreted as temperature, similarities can be drawn between the governing equations for chemo-poro-elasticity and the equations of the thermo-poro-elastic logic implemented in FLAC.

The logic has been applied to an elastic oedometric test case for which an analytical solution exists, and comparison of numerical and analytical solutions for the verification test show an excellent match. The quality of the match indicated that, for the case investigated, the impact of volumetric strain and pore pressure on osmotic pressure can be ignored, as had been assumed in this work. A borehole problem has been considered, with a Mohr-Coulomb constitutive model adopted for the shale. A parametric study has been conducted to investigate the effect of reflec-tion coefficient and coupling coefficient on borehole stability for a case study. The results of the numerical experiments show that an increase in reflection coefficient reduces the pore and osmotic pressure in the formation, and has a positive impact on borehole stability (i.e. the extent of yielding is reduced). Also, the effect of coupling coefficients and , for = and for giv-en R, has little effect on problem response for the conditions investigated in the case study.

REFERENCES

Bunger, A.P. 2010. The Mandel-Cryer Effect in Chemoporoelasticity. Int. J. Numer. Anal. Meth. Geomech. 34, 1479-1511.

Coussy, O. 2004. Poromechanics. New York: John Wiley & Sons. Ekbote, S. & Abousleiman, Y. 2003. Poromechanics Stability Analyses for Inclined Wellbores under

Non-Isothermal Conditions in Chemically Active Formations. In Proceedings, 16th ASCE Engineering Mechanics Conference (University of Washington, Seattle, July 2003).

Ghassemi, A. & Diek, A. 2003. Linear chemo-poroelasticity for swelling shales: theory and application. J. Petrol. Sci. Eng. 38, 199-212.

Ghassemi, A., Tao, Q. & Diek, A. 2009. Influence of coupled and chemo-poro-thermoelastic processes on pore pressure and stress distributions around a wellbore in swelling shale. J. Petrol. Sci. Eng. 67, 57-64.

Heidug, W. & Wong, S.W. 1996. Hydration Swelling of Water-Absorbing Rocks: A Constitutive Model. Int. J. Rock Mech. Min. Sci. 20, 403-430.

Itasca Consulting Group, Inc. 2011. FLAC – Fast Lagrangian Analysis of Continua, Ver. 7.0. Minneap-olis: Itasca.

Mody, F.K. & Hale, A.H. 1992. A Borehole Stability Model to Couple the Mechanics and Chemistry of Drilling Fluid Shale Interactions. In Proceedings, IADC/SPE Drilling Conference (Amsterdam, 1993), Paper No. SPE25728.

Roshan, H. & Fahad, M. 2012. Chemo-poroplastic analysis of a borehole drilled in a naturally fracture chemically active formation. Int. J. Rock Mech. Min. Sci. 52, 82-91.

Roshan, H. & Rahman, S.S. 2011. 3D borehole model for evaluation of wellbore instabilities in underbal-anced drilling. In Proceedings SPE EUROPEC/EAGE Annual Conference and Exhibition (Vienna, 2011), Paper No. SPE143320.

Sarout, J. & Detournay, E. 2011. Chemoporoelastic analysis and experimental validation of the pore pres-sure transmission test for reactive shales. Int. J. Rock Mech. Min. Sci., 48, 759-772.

Sherwood, J.D. 1994. A Model of Hindered Solute Transport in a Poroelastic Shale. Proc. Roy. Soc. Lon-don A, 445, 679-692.

Steiger, R.P. & Leung, P.K. 1992. Quantitative Determination of the Mechanical Properties of Shales. SPE Drill. Eng. 7(3), 181-185.