When enhanced oil recovery procedures were introduced to the industry, thermal stresses in and around the wellbore were discovered and after that the effect of coupling the thermal and mechanical stresses came to the investigation to better forecast the behavior of casing and cement during different thermal pilots.
6.4 Modeling
In this project, the stresses in the cement are calculated assuming that steel, cement, and rock are thermo-elastic materials and that the steel/cement interface and the cement/rock interface are either fully bounded or unbounded. Finally, the analysis presented here, assumes that the cement is under no internal stress after setting; only the variations of pressure, stress, or temperature that occur once the cement is set are considered.
The geometry of the problem is axisymmetric with the axis of symmetry being the wellbore axes, allowing the use of cylindrical coordinates r, θ, and z. The simplest situation is when the boundary and initial conditions (wellbore pressure, far-field state of stress, wellbore, and far-field temperature) are independent of θ. The variables of interest are then the radial displacement, radial stress, σr; tangential stress, σt; axial stress, σx; the shear stress τ, and temperature, T.
We model the wellbore with casing, cement, and the formation in one play. (Play, means the acting system of casing, cement, and formation.) This play can be very big or small in diameter or thickness. The hydrostatic mud pressure from the wellbore is the inner pressure in this play, and the horizontal in-situ stress acting on the outer boundary of cement is the outer pressure.
In this literature, we calculate all the rock mechanics parameters in Cartesian coordinate. There is no unit conversion in this literature regarding to this matter. The conversion has been done before the putting data sets in the FE simulator.
We ran some limited models just to investigate the effect of solid rock mechanics such as Young’s modulus, Poison’s ratio, and the effects of temperature, eccentricity, pressure differential and casing or cement thickness.
In these literatures22-24different types of cement were studied and applied to different wells. Table 6.1 to Table 6.4 governs the required data. Based on the type of additives and wellbore pressure and temperature, the solid mechanics behavior of the set cement changed and some of the cements failed in either internally or from boundaries. In this study, we are using the same cement characteristics in our modeling to monitor the behavior of these cements in various pressure and temperature conditions.
Table 6.1− Mechanical and Thermal Properties of Different Constituents 22-24
Table 6.2− Cement Property Definition Cement
Type Property Description
Slurry-1 Class G 22 Slurry-2 Hybrid 22 Slurry-3 Latex-Slurry 22 Slurry-4 Foamed Slurry-I 22 Slurry-5 Foamed Slurry-II 22
Slurry-6 Class A with 15%Multi Purpose Additive, .4%Extender Sodium Silicate, 0.45%
CMHEC, .25% SNSC 23
Slurry-7 Class A with 0.36gr Extender Sodium Silicate, 0.45% CMHEC 23 Slurry-8 Class A with 6%Extender Bentonite 23
Slurry-9 Class A with 0.2% CMHEC, 0.2% SNSC 23
Slurry-10 Class A with 50%Multi Purpose Additive, 0.15% CMHEC, 0.15% SNSC 23 Slurry-11 Class A with 15%Multi Purpose Additive, 0.25%Extender Sodium Silicate,
0.8% CMHEC, 0.45% SNSC 23 Slurry-12 Conventional neat Cement 24 Slurry-13 Flexible Cement 24
Slurry-14 Flexible Cement (different from Slurry-13) 24 Slurry-15 Flexible and Expanding(based on Slurry-13) 24
Slurry-16 Flexible and Expanding (based on Slurry-13more expandable than Slurry-15) 24 Slurry-17 Flexible and Expanding (based on Slurry-13more expandable than Slurry-16) 24 Slurry-18 Atmospheric foamed cement (30% quality) with expanding agent 24
Table 6.3− Geometry, Formation, and Fluids Data at Intermediate Casing23 Geometry Information Formation and fluid Information
Hole Size, inch. 12.25 Mud Density, ppg 14.5
Casing OD, inch. 9.875 Cement, ppg 15.5
Displacement Fluid, ppg
(Casing) 14.5
Table 6.4− Pressure/Temperature in Intermediate Casing During Well Events23 Casing ID, inch. 8.539
Displacement Fluid, ppg (Production Liner)
17.1 Casing Grade Q125, 67.5 lb/ft Completion Fluid, ppg 8.4
16000
Formation Type limestone Depth (TVD), ft
Previous Shoe
(TVD), ft 12000 Formation Vertical Stress Gradient,
psi/ft 0.94
Eccentricity, % 10 Formation Pore Pressure, psi/ft 0.46 Formation Fracture Pressure, psi/ft 0.62
Event Remarks T ′ s
inside the casing 350 50 12064-9280 12064-9920 Completion
Completion fluid
inside the casing 350 60 6989-9280 6989-9920
Pressure Testing
Completion fluid
inside the casing 350 60 6989-9280 6989-9920
Production Through Liner
Hydrocarbon
Production 350 0 15392-9280 14227-9920
The most reliable equation for calculating fracture pressure in the gulf coast is Ben Eaton’s equation and can be written as Eq. 6.1.
p p ob
H − p + p
= − )( )
(1 σ ν
σ ν ,...……… (6.1)
The fracture gradient depends upon the location, depth, and formation. (In off-shore wells, the quantity of fracture gradient varies with the depth of water)
For depth of 16000 ft at the gulf coast margin in Texas, Poisson’s ratio is equal to 0.25 in the calculations on Table 6.4.
Eq. 6.1 explains the upper limit of the inside wellbore pressure. If the pressure inside the wellbore exceeds this amount, it means that we pass the fracture pressure and we are fracturing our system ourselves, which is not desirable. As a consequence, we are endangering the whole structure to gas migration phenomena, and following that, the structure will collapse.
CHAPTER VII RESULTS
Results that had an extreme effect on the case study reveal the effects of Young’s modulus, Poisson’s ratio, casing thickness, cement thickness, casing eccentricity, and temperature effect on each case.
Fig. 7.1 shows the model that we made. Notations on the figure are described as follows:
Area A, the system is under constraint load to lateral sides which means that the system can not move axially on the y-axis but it is free to move on the x-axis. Area B shows the type of analysis, which is nodal, and some governing data regarding the type of solution, like central processor unit, CPU, time. Area C is an index to the colors used in the analysis. These color codes are based on the intensity of the stress in each region. The number beside each color shows the magnitude of the stress at the area. The units in each figure are consistent with the input data units.
The correct behavior of the system can be found in the areas above or below the constraint-acting area. Since Young’s modulus is the ratio of stress to strain, this number is a good basis for the flexibility of each medium or solid. If this parameter is too low, the medium will act plastically, like unconsolidated sand stone with high water saturation, and result in system instability. But if this number is low, the range of stresses which can be handled by the system will increase. The higher the Young’s modulus, the stiffer the mass is. As it is shown in Fig. 7.1 (applying input data from Table 7.1) when the Young’s modulus of cement is very high, in this case 10 times more than Young’s modulus of casing, almost all the force is being applied to the casing, all deformation will be in the casing, and the cement will remain intact. In this situation, because the cement is acting more flexible than casing, it may cause casing to move and consequently increase the friction of the drill string in the wellbore resulting to early corrosion. As Fig.
7.1 shows, the strain in the casing is 20 times more than cement. It means that the casing failure is more possible than cement failure and as a result of casing failure, we have casing/cement boundary fail or de-bonding.
The effect of Poisson’s ratio is significant. Since Poisson’s ratio is the ratio of transverse contraction strain to longitudinal extension strain in the direction of the applied
force. In this study, since the model is investigating in 2D form, Poisson’s ratio plays an extreme role in mass deformation. The plane-strain assumption based on no displacement along the wellbore direction is recommended for field use, so increasing transversal strain should increase in Poisson’s ratio or vice versa.
TABLE 7.1− Input Data Required for Different Cases
Case
Event completion completion completion pressure
test completion completion completion
Slurry type 1 1 1 1 1 1 1
pressure 9280 9280 9280 10000 9280 9280 12064
Pore pressure 12064 12064 12064 15000 12064 12064 9280
Temperature 350 350 350 350 400 350 350
A
B
C
Fig. 7.1− Strain deformation while changing Young’s modulus of cement (Ecement=10×Ecasing)
Fig. 7.2 (applying input data from Table 7.1) will resolve this effect more sensibly. In these cases, since the steel Poisson’s Ratio is hard to change with respect to cement, including additives in the cement will keep this parameter in a reasonable range in any type of well completion.
If the two runs on Fig 7.2 and Fig 7.3 (applying input data from Table 7.1) are compared with each other, the result is interesting. The cement had some deformation, laterally and radially, because of the difference between its Young’s modulus and the formation. The effect of Poisson’s ratio as shown is not significant. (The major deformation is because the low Young’s modulus does not change Poisson’s ratio but the range of Poisson’s ratio has been changed by a factor of 40 in these two figures.)
The effect of pressure is very important in underbalanced drilling. The effect of pressure difference between the inside and outer boundary will change the results significantly and may cause expensive damage and permanent deformation in the system.
Fig. 7.4 (applying input data from Table 7.1) shows the effect of a 5000-psi difference between the inner and outer boundary conditions of the system. This case runs under the same conditions as earlier ones. To ignore the effect of probable inside casing damage, we assume that the entire casing is fixed and the pressure difference causes no expansion or contraction on casing. Because both Young’s modulus and Poisson’s ratio are very pressure dependent, the behavior of pressure in such system should follow the same as Young’s modulus and Poisson’s ratio. Referring to Poisson’s ratio by definition, we can conclude that the strain distribution in case of having differential pressure on the system should follow exactly the same behavior as Poisson’s ratio. This can be proved comparing Fig. 7.2, and Fig. 7.4.
Temperature has a huge effect on the system. Since the solid mechanic properties of the combined masses are strongly interconnected with temperature, its variation will cause some failure in the material composition and increase corrosion in the presence of corrosive fluids. The effect of temperature has been denoted in Fig. 7.5 (applying input data from Table 7.1). In this figure, the Area D cement-casing edge has been put under another 500F thermal stress. Obviously, this imposed temperature increases the chance of failure in that section.
Eccentricity has its own case study. Based on type of cement, casing, inner and outer pressures, and enforced temperature, this parameter becomes important. Fig. 7.6 (applying input data from Table 7.1) denotes the relative behavior of eccentricity of the casing. If we couple the effect of thermal stress and eccentricity, we come up to a severe scenario for cement failure. In this situation, the chance of cement/casing Debonding will be the same all over the boundary regardless of cement thickness.
Casing thickness and cement thickness have been investigated in Fig. 7.7. The relevant data for this run are located in Table 7.1. In this case where the ratio of casing thickness to cement thickness is more than 2, the dominant stress concentration will be focused on the layer with higher the Young’s modulus, because the more flexible matter will transmit the stress or strain to the adjacent matter. In such cases, there may be some
de-bonding may appear in the margin between two different materials as a result of corrosion in the long-term wellbore life, especially in HPHT cases.
Fig. 7.2− Strain distribution while the Poisson’s ratio is 0.01 (Strain is low in casing, very high in casing/cement boundary and lower in
cement/formation boundary)
Fig. 7.3− Strain distribution while the Poisson’s ratio is 0.4 (40 times more than Fig. 7.2.
since quantity of Poisson’s ratio of two adjacent matters is very close to each other, the strain distribution is uniform in each medium)
Fig. 7.4− Effect of high pressure difference between inside the casing and in the outer boundary of cement (in this case, the pressure difference is 5000 psi)
D
Fig. 7.5− Effect of temperature on the edge strain (extra temperature added on the cement-casing boundary in area D)
Fig. 7.6− Strain distribution in eccentric casing (Higher cement failure chance in thicker cement side)
Fig. 7.7− Stress distribution while changing casing and cement thickness (The behavior is quite uniform and chance of failure is low)
In the Fig. 7.8 the shear stress in X-Y coordinate has been sketched for the system covering data from Table 7.2.
Table 7.2− Input Data for Investigating Shear-Stress in Slurry-1
Event Slurry Type Casing ID, in Casing OD, in Casing Young’s Modulus, psi Casing Poisson’s Ratio Cement OD, in Cement Young’s Modulus, psi Cement Poisson’s Ratio PBH, psi Pp, psi Temperature, 0F
completion 1 8.539 9.875 30e6 0.27 12.25 1.2e6 .1 6989 9280 350
Fig. 7.8− Shear-stress distribution in x-y coordinates
In this scenario, four sections behave exactly the same but with the opposite sign of force as shown in Area C (Tension or compression).The primary assumed element (quadratic) is shown in Fig. 7.9. In this sketch, the sides of the element are exactly parallel and equal to each other.
x y
Fig. 7.9− Primary element model (quadratic, eight nodes)
The trend of shear stress in each quarter of the casing-cement system is maximized in the half. (Half; at 450 we encounter the maximum shear stress) But this shear stress
quantity is positive in the first and third quarters and negative but equal in the second and forth quarters. Fig. 7.10 depicts the direction of maximum shear and the reflection of primary element to that in the first quarter. Those elements on the second and fourth quarter behave as shown in Fig. 7.11.
y
x
Fig. 7.10− Element verification in 1st and 3rd quarters
y
x
Fig. 7.11− Element verification in 2nd and 4th quarters
To investigate the cement or casing behavior, we need to know the shear stress of each. According to Fig 7.8 if the cement passes the limit of maximum shear, it will fail.
The rock properties of cement will change depending on the its make up. In this simulation, since the meshes are not too fine, the final shear stress behavior of all the slurries (tabulated in Table 6.1) in one set of inside and outside pressures will be the same. But since the shear stress is strongly related to cement aggregation, the pressure won’t be the same, and some slurries will fail. (Slurries in set cement form)
If we force another set of pressures inside and outside the system, Fig. 7.12 (sketch based on Table 7.3 data) the reaction of the entire system will change with new applied pressures. Cement behavior under this circumstance is strongly related to its aggregation and the induced pressures.
Table 7.3− Input Data for Shear-Stress Test on Slurry-9
Event Slurry Type Casing ID, in Casing OD, in Casing Young’s Modulus, psi Casing Poisson’s Ratio Cement OD, in Cement Young’s Modulus, psi Cement Poisson’s Ratio PBH, psi Pp, psi Temperature, 0F
completion 9 8.5 39 9.8
75 30e6 0.27 12.2
5 .95e6 .18 9280 15392 350
Fig. 7.12− Shear-stress profile in case of forcing higher outside pressure
Because the same cement is under different imposed pressure in the same well, the chance of failure increases. The failures are mostly casing parting, cement-formation parting, and cement breakage.
According to Eq. 7.1 shear-stress changes with area (A) and thickness, moment of inertia of the passive surface and shear force (F). In this case, the area and thicknesses remain constant, so shear-stress is just changing under applied force (F), which is why whenever the force goes up, the shear stress increases linearly.
A F
A Δ
= Δ
→
lim
Δ 0τ , ……….……… (7.1)
The acting force and cross-sectional area are shown in Fig. 7.13.
x
ΔA F
Fig. 7.13− Active force and the cross sectional area
The deformation caused by shear-stress is called shear-strain. The ratio of shear-stress to shear-strain based on Hook’s law is called shear module of elasticity; G. Shear-strain is depicted in Fig. 7.14.
Fig. 7.14− Element of material with applied shear-stress,τ , and shear-strain, γ
The behavior of shear-stress is dependent on the amount of pressure acting on the system. Since no shear-stress data are available for the cements that have already been modeled, we cannot conclude any form of deformation for those systems. But since the cement is very shear dependent and will collapse under high shear-stress, we can infer that the system may fail if it goes under frequent pressure testing.
The following cases illustrate casing-cement interaction with different cement solid properties under different sets of imposed pressures. Table 7.4 governs data needed for Fig. 7.15. If we compare the results in Fig. 7.15 and Fig. 7.16 (ignoring the change in Young’s modulus and Poisson’s ratio, Table 7.5), we can conclude that both shear-stress and shear-strain are related to pressure changes rather than changes in cement properties.
It does not mean that shear-stress or shear-strain are independent from the material properties of solids, but it means that these parameters (shear-stress and shear-strain) mostly take the advantage of pressure variations.
Table 7.4− Input Data for Investigating Shear-Strain in Slurry-2
Event Slurry Type Casing ID, in Casing OD, in Casing Young’s Modulus, psi Casing Poisson’s Ratio Cement OD, in Cement Young’s Modulus, psi Cement Poisson’s Ratio PBH, psi Pp, psi Temperature, 0F
Pressure
Testing 2 8.539 9.875 30e6 0.27 12.25 .174e6 .17 9280 15392 350
Fig. 7.15− Shear-strain test in pressure test (high pressure)
Table 7.5− Input Data for Investigating Shear-Strain in Slurry-4
Event Slurry Type Casing ID, in Casing OD, in Casing Young’s Modulus, psi Casing Poisson’s Ratio Cement OD, in Cement Young’s Modulus, psi Cement Poisson’s Ratio PBH, psi Pp, psi Temperature, 0F
Pressure
Testing 4 8.539 9.875 30e6 0.27 12.25 .571e6 .23 6989 9280 350
Fig. 7.16− Shear-strain test in pressure test (low pressure)
The von-Mises failure criterion is known as the average resultant stress criterion.
Based on this application, whenever the two horizontal stresses are equal, this criterion react the same as other available criteria.(like Drucker-Prager) In Fig. 7.17 the reaction of the system based on data labeled on Table 7.6 under von-Mises criterion is depicted.
Since everything is symmetric, the results are in symmetric form. As Fig. 7.18 shows the minimum and maximum acting pressures are more than those we estimated.
Table 7.6− Input Data for Investigating Stress in Slurry-5
Event Slurry Type Casing ID, in Casing OD, in Casing Young’s Modulus, psi Casing Poisson’s Ratio Cement OD, in Cement Young’s Modulus, psi Cement Poisson’s Ratio PBH, psi Pp, psi Temperature, 0F
Pressure
Testing 5 8.539 9.875 30e6 0.27 12.25 .015e6 .12 6989 9280 350
Fig. 7.17− Stress profile based on von-Mises criterion in pressure test
Fig. 7.18 and Fig. 7.19 (based on data in Table 7.7) show clearly the reaction of the system to higher inside pressure. Comparing Fig. 7.19 with Fig. 7.16 shows that if the dominant pressure forces changes (switch inside to outside), the system still reacts the same. This same reaction is mostly because of the material properties of the system attaching to each other. That is the reason we said that in previous runs that the shear-
stress and shear-strain are not just reacting because of induced pressure, but also because of the material properties of the system.
Table 7.7− Input Data for Investigating Shear-Stress in Slurry-13
Event Slurry Type Casing ID, in Casing OD, in Casing Young’s Modulus, psi Casing Poisson’s Ratio Cement OD, in Cement Young’s Modulus, psi Cement Poisson’s Ratio PBH, psi Pp, psi Temperature, 0F
Hydrocarbon
Production 13 8.539 9.875 30e6 0.27 12.25 .094e6 .2 15392 9280 350
Fig 7.18− Shear-stress profile for slurry-13 under hydrocarbon production
Fig 7.19− Shear-strain profile for slurry-13 under hydrocarbon production
CHAPTER VIII CONCLUSIONS
This project has shown that:1- Changing temperature can cause unexpected behavior in the cement sheath, which leads to failure in the casing/cement bond.
2- In the case of eccentric casing, the cement/casing bond will break radially, mostly on the side of the thicker cement sheath.
2- In the case of eccentric casing, the cement/casing bond will break radially, mostly on the side of the thicker cement sheath.