Pipeline construction technology through horizontal directional drilling through complex terrain such as rivers has been applied on a large scale. In order to ensure the integrity of the pipeline, mechanical scratches are generally prevented by tamping the installation casing. Accurately obtaining the stress response of the casing during the tamping process is an important part of evaluating the safety of the casing structure. The key is to explore the complex casing-soil coupling mechanism and formulate a reliable numerical analysis strategy. In the traditional numerical model, the fine mesh generation and hard contact algorithm are usually used, which leads to the high computational cost of the model, and the simulation efficiency is difficult to meet the engineering application requirements. To address this issue, a spatiotemporal-evolution nonlinear pipe–soil interaction model (STN-PSIM) is proposed. In this model, the soil constraint is equivalent to the end and side mechanical elements that change with the depth of the soil and the nature of the formation, and the stress distribution in the process of casing ramming is solved based on statics. The results show that the model accurately restores the casing ramming process, and the calculated results are in good agreement with the experimental data. Compared with the traditional large deformation finite element method, the model avoids soil meshing and significantly improves the computational efficiency under the premise of ensuring accuracy. The research results can provide scientific basis for pipe selection, ramming force configuration and casing safety evaluation.