Abstract: The horizontal oil and gas well pipe rod is simplified as a viscoelastic cylindrical pipe model with elastic constraints. The fractional derivative viscoelastic theory is used to describe the constitutive relationship of the pipe rod. Combined with the theory of elastic mechanics, the equilibrium equations of compressible viscoelastic pipe rod are established, and the analytical solutions of viscoelastic stress and displacement of compressible pipe rod are obtained. The numerical results show that the larger the order of the fractional derivative, the larger the initial stress at the inner wall of the corresponding pipe rod, the larger the radial and circumferential stress at the inner wall of the cylindrical pipe, the larger the rising section of the corresponding vertical stress, and the larger the stable value of the vertical stress of the pipe rod. When the the Poisson's ratio of the outer casing increases, the radial displacement and stress will be smaller. The influence of Poisson's ratio on the circumferential stress is not obvious, but the influence on radial stress is more significant. The elastic modulus of casing has little effect on the radial displacement. The larger the model constant ratio, the larger the radial and circumferential stresses. The thickness of the outer casing has little influence on the circumferential stress at the inner wall of the column pipe. When the thickness of the outer casing increases, the radial stress will also increase, and the vertical stress value will decrease.