Structural integrity assessments are vital for ensuring the safety and efficiency of oil and gas wells, especially in sour service applications. The casings used in drilling operations are critical as mechanical barriers against leaks among different well-construction components. However, their susceptibility to environment-assisted crack growth, like sulfide stress cracking (SSC), presents challenges for casing mechanical integrity management. Conventional analytical methods are quick but can be overly conservative in material selection. Recently, multiphysics modelling of fracture has emerged as an accurate simulation approach, leveraging tools such as hydrogen diffusion models, fracture mechanics, and finite element analysis. In this work, a coupled deformation-diffusion phase-field finite element framework is used to model SSC nucleation and growth in a sour environment. The multiphysics model employs coupling between structural deformation, hydrogen diffusion due to H2S exposure, and fracture processes to simulate SSC. The numerical results show good agreement with the experimental data for different levels of H2S exposure. A numerical study is also conducted to study SSC nucleation and growth in pre-notched mini-pipe subjected to internal pressure and H2S exposure. The findings of this investigation provide valuable insights into the effectiveness of a coupled phase-field approach to study the combined role of stresses and through-wall hydrogen gradients on pipe failure.