The classical Lucas-Washburn (L-W) capillary-driven model calculates capillary pressure using the Young-Laplace equation, which leads to a deviation from the real values for a pipe with small diameter. In this work we use Tolman length to improve the Young-Laplace equation, and propose an improved L-W model. Moreover, circular pipes with rough and variable sections are considered rather than the circular pipe with a constant diameter. Relation between time and wetting fluid injection length in the circular pipe with variable sections is established. The mathematical model is described as a second-order nonlinear ordinary differential equation, which cannot be solved analytically, and thus an efficient numerical method is developed. A concrete pipe with variable cross-section is selected, and the relation between water length and time is calulated with numerical simulation. Numerical results are analyzed, and the effectiveness of Tolman length is verified. It shows that the local shrinkage of circular pipe changes the states of water motion significantly, and there are three kinds of movement modes. The local expansion of circular pipe changes slowly the state of seepage and water absorption, and only one single movement mode takes place.