Calibration of the interface model and model parameter setting

To calibrate the parameters of the interface model, a finite element analysis of a laboratory experiment, where a well specimen is subjected to tensile loading (Figure 5-5a), is carried out. The simulated axial strain development of the well specimen during tensile loading is compared with the experimental data. The details of the experiments are given in Chapter 6.

(a)

(b)

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loading apparatus; (b) a cross section of the well specimen.

The cross-section of the well specimen is illustrated in Figure 5-5b. The specimen consists of inner casing, cement sheath and outer casing. Fibre optic cables for distributed axial strain monitoring are embedded in the cement, whereas strain gauges are attached the inner and outer casings.

Figure 5-6 shows the axi-symmetric finite element model used in this study. The inner casing, cement and outer casing are modelled with the eight-node biquadratic displacement element. The inner and outer casings are discretized into 600 elements each whereas the cement is discretized into 12,000 elements. To apply a tensile load, the displacements of the bottom nodes of the inner and outer casing are fixed and a distributed load is applied on the top nodes of the outer casing in the vertical direction.

Figure 5-6 The axi-symmetric finite element model of the well specimen (left) and the model mesh (right).

The inner and outer casings are modelled isotropic linear elastic. The Young’s modulus and Poisson’s ratio are set to 200 GPa and 0.26, respectively, which correspond to the typical values of A36 steel which comprises the casings. The cement is modelled isotropic linear elastic with the Mohr-Coulomb yield surface. The Young’s modulus and Poisson’s ratio of the cement are set to 8.3 GPa and 0.10, which are taken from the literature (Bosma et al. 1999). The values of the internal friction angle and cohesion of the Mohr-Coulomb yield surface for the cement are also taken from the literature (Bosma et al. 1999) and are set to 17.1o _{and 21.6 MPa,}

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respectively. The dilation angle is assumed to be 0o_{. For the casing-cement interface, the values}

of the friction coefficient and cohesion are respectively set to 0.8 (i.e., friction angle of 38.7o₎

and 3.0 MPa. These values are based on the experimental result of the pushout test of a steel rod embedded in cylindrical cement sheath (Yoneda et al. 2014). The remaining parameter is the ultimate elastic interface displacement and a typical value for the casing-cement interface could not be obtained from the literature. Hence, this value is calibrated to match the simulation with the experimental data. The match between the simulation and experiment is judged visually and no proper optimisation such as the least square method is performed.

The model calibration result is shown in Figure 5-7. The numbers in the figures indicate the axial load increments (1 kips = 4.45 kN). It is found that a good match between the simulation and experiment is obtained with the value of ultimate elastic interface displacement = 0.5 mm. To validate the calibration result, an analytical solution for the casing-cement shaft friction problem is provided in Chapter 6 (Section 6.4.2.), which shows that the choice of 0.5 mm is appropriate as the numerical and analytical results match satisfactorily, including the gradient of axial strain at the top and bottom of the specimen, at small load levels (< 125 kips) where the cement is still elastic. Once the cement develops plastic strains after mid-load levels (> 175 kips), the gradient of axial strain increases significantly. The sharp strain gradient after the cement yield is dependent on the plastic parameters of the cement (i.e., friction angle, dilation angle and cohesion) and hence it could not be matched by changing solely the value of the ultimate elastic interface displacement.

For the cement-formation interface of the reservoir compaction simulation, it is assumed that the interface friction coefficient is identical to that of the underlying formation. The friction coefficient of the overburden layer is 0.67 (i.e., friction angle of 33.9o_{) whereas that of the MH}

reservoir and underburden layers is 0.63 (i.e., friction angle of 32.3o_{). As the difference is small,}

the mean value of 0.65 is used for the entire cement-formation interface. For the interface cohesion at the cement-formation interface, it is assumed negligible as soil particles of the unconsolidated formation do not resist friction at zero interface confining pressure, which is experimentally validated in the literature (Yoneda et al. 2014). The value of the ultimate elastic interface displacement is set to 0.25 mm for the cement-formation interface. This is determined by varying the value of ultimate elastic interface displacement between 0.25 mm and 2.5 mm and carrying out the reservoir compaction simulation which is presented in the following sections. Results show negligible differences in the development of stresses and strains of the casing and cement during reservoir compaction. Therefore, the value is set to 0.25 mm. To support this, an experimental study by Uesugi et al. (1990), where a sand specimen prepared

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inside a stack of rectangular frames is sheared against a mortar plate placed beneath the sand while vertical confining pressure is maintained, shows that the value of ultimate elastic interface displacement between the sand and mortar is approximately 0.3 mm.

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simulation and experiment: (a) inner casing; (b) cement; (c) outer casing.

It is noted that the level of interface confining pressure between the laboratory experiment (Figure 5-5) (atmospheric pressure) and the actual wellbore in the Nankai Trough (~10 MPa) is significantly different. Although a better calibration of the friction model parameters could have been performed if the experiment had been conducted under the actual confining pressure conditions, the application of ~10 MPa confining pressure over the 3 m-long specimen was not feasible in the laboratory.

5.2.6. Initial conditions

For the Nankai Trough case, the initial vertical stress distribution of the formation is derived from the in suit density measurement at the site (Suzuki et al. 2015). The initial void ratio distribution is also obtained from the same in situ density measurement. For the initial pore pressure distribution, the hydrostatic pore pressure distribution with the seawater density of 1.027 g/cm3 _{is employed.}

Figure 5-8 Initial horizontal effective stress distributions of the formation.

Two different initial horizontal effective stress distributions are employed as shown in Figure 5-8. The overconsolidated distribution is calculated via Equation 5-9:

𝜎′ℎ= (1 − sin𝜙′)OCRsin𝜙

′

𝜎′𝑣 (5-9)

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angle, OCR = overconsolidation ratio. This formula is often employed in the soil mechanics. The OCR values of the overburden layer is derived from triaxial test data of formation core samples retrieved from the Nankai Trough (Nishio et al. 2011). Hence, the overconsolidated formation case is more representative of the actual Nankai Trough formation. For the normally consolidated case, the initial horizontal effective stress is calculated via 𝜎_ℎ′ _{= 0.4𝜎}

𝑣′. The effect

of initial lateral pressure will be examined in Section 5.3.6.