عنوان مقاله [English]
نویسندگان [English]چکیده [English]
Young’s modulus measured as the slope of a stress-strain curve under static loading conditions (Es) in the lab is an essential rock mechanical parameter for geomechanical analyses of oil wells. Examples of these analyses are wellbore stability analysis, estimation of the in-situ stresses, and the reservoir compaction survey. However, Es is obtained by destructive laboratory tests on selected core samples along the well length. Therefore, information on the value of Es along the well length is often discontinuous and limited to a cross well with a core. On the other hand, based on the theory of elasticity, well-known equations are available to calculate Young’s modulus under dynamic (compressional and shear wave) loading conditions which is the dynamic Young’s modulus (Ed). Nevertheless, Ed for intact core specimens is very often two or three times or more larger than Es. This is partly because in case of a dynamic loading strain, the amplitude is 10-6 or 10-7, while in the static moduli strain, amplitude is typically 10−2–10−3. Static moduli, measured as the slopes of the stress–strain curves, differ from small strain amplitude dynamic (elastic) moduli because of plasticity or nonlinear effects. Also, porosity and micro-cracks in rock core specimens affect this phenomenon. Accordingly, many attempts have been made to predict Es based on other nondestructive parameters namely compressional and shear wave velocities (Vp and Vs) in the lab. Fortunately, geophysical logs in many hydrocarbon reservoirs provide Vp and Vs data continuously along the well length. Therefore, it is possible to calculate Ed continuously in the well. For this reason empirical equations have been developed to estimate Es based on Ed along the well length.
Furthermore, a correlation between Ed and Es at in-situ conditions is more difficult than in the lab. This is because Vp and Vs measurements increase by increasing in-situ confining stresses. This is in turn because confining stresses reduce the anisotropy elements such as porosity and micro-cracks. As a result, Vp and Vs Ed often will increase with the burial depth. Similarly, laboratory experiments indicate that stress-strain measurements on rock core specimens under the static loading and triaxial confining stresses (similar to the well depth) will increase Es with an increase in the confining stresses.
In this research, laboratory experiments were carried out on limestone rock core specimens of Sarvak Formation obtained from an oil well in the South West of Iran. The specimens were placed in a cell under confinement. Compressional and shear wave velocities at different confining stresses were measured. Experiments were accomplished in the dry conditions up to a maximum confining pressure of 50 MPa. Simultaneously, the values of the axial load and axial strain were recorded. It was noticed that with an increase in confining stresses, Vp and Vs will increase. Likewise, at lower confining stresses, Vp and Vs increase exponentially while after a critical confining stress of 15 MPa, exponential equation will turn to linear. Constants of the linear and exponential equations for Sarvak formation were extracted with excellent accuracy.
Based on these measurements, Ed and Es were calculated at different levels of confining stresses. It was observed that with an increase in confinement, the ratio of Ed/Es will decrease and approach to unity at higher confinements. This means that with an increase in confining stresses, Es will increase faster (compared to Ed). According to the findings of this research, a correlation between Ed and Es should be made with extreme care to account for the impact of the confining stress at any depth of interest. This is very often ignored throughout the well length. Finally, based on laboratory experiments, an empirical equation was developed to predict Es from Ed at different confining stresses.